CONDENSED  LIST. 

AP-  means  Within  2%. 


Physical  Quantities,  Relations, 
Dimensional         Formulas, 

etc.,  p.  18-26 

lengths:  complete  tables,  p.  30-34 
1  mil  =  0.025  40  mm.     Ap.  *40 
1  mm.  =39.37  mils.     Ap  40 
1  cm.  =0.393  7  inch.     Ap.  Vi0 
1  inch  =  2. 540  centimeters.    Ap.  10/i 
1  foot  =0.304  8  meter.    Ap.  3/l0 
1  yard  =  0.914 4 meter.  Ap^Hoor^i 
1  meter  =  39. 37  inches.    Ap.  40 

=  3.281  feet.    Ap.  1% 

"      =1.094  yards.    Ap.  iyw 

1  kilometer  =  3  281.  feet.     Ap.  MX 

10000 

"      =1  094.  yards.     Ap.  1  100 
"      =0.621  4  mile.    Ap.  ^ 
1  mile  =5  280.  feet.     Ap.  5  300 
"      =1  760.  yards.    Ap.  %  X  1  000 
"      =1.609  km.    Ap.  add  <K0 
1  knot  =  1.853  kilometers.     Ap.  *% 

44  =1.152  miles.  Ap.addiA 
Inches  in  fractions,  decimals,  milli- 
meters and  feet,  p.  35-38 
Digit  conv.  tables  (1  to  100),  p.  39-40 
Surfaces:  complete  tables, p.  41-44 
1  circular  mil  =  0.000  506  7  sq,  mm. 

Ap.  H-s-1  000 
1  sq.  mm.  =  l  974.  circ.  mils.     Ap. 

2000 

1  sq.  cm.  =0.155 0  sq.  inch.    Ap.  ^.3 
1  sq.  inch  =  1  273  240.  circ.  mils 

=  6. 452  sq.  cm.  Ap.  §1A 
1  sq.  ft.  =0.092  90  sq.m.  Ap.H£2-5- 10 
1  sq.  yd.  =0.836  1  sq.  meter.  Ap.  % 
1  sq.  m.  =  10.76  sq.  ft.  Ap.  %  x  ,0 
"  =1.196  sq.  yds.  Ap.  addVs 
1  acre  =43  560.  sq.  feet 

44      =0.4047  hectare.    Ap.  ^ 
1  hectare  =  2.471  acres.    Ap.  1% 
-      "         =0.003  861  sq.  mile 
lsq.km.=  247.1  acres.  Ap.MXIOOO 
=  0. 386  Isq. mile.     Ap.1';^ 
1  sq. mile  =640.  acres 

=  2.590sq.km.  Ap.26-  10 
Digit  conversion  tables,  p.  14 

To  1  umes :  complete  tables  p .  46-55 
1  cb.  cm.  =0.061  02  cb.  in.  Ap.  %oo 
1  cb.  in.  =  16. 39  cb.  cm.    Ap.  10% 
1  pint=  28.875  cb.  in.     Ap.2AX100 
44      =0.473  2  liter.     Ap.  Vsi  X  10 
1  quart  =0.946  4  liter.    Ap.  sub.  Vao 
lliter  =  61.02  cb.  inches.    Ap.  60 
"     =1.057  quarts.    Ap.  add  ^20 
"    =  0.035  31  cb.  ft.    Ap.%-^100 
1  gal.  (U.  S.)  =231.  cb.  inches 

=  3.785  liters.    Ap.^XlO 

"          =0.133  7  cb.  ft.  Ap.  % •*•  10 

"      (Brit.)  =4.546  liters.    Ap.  4^ 

1  cb.  ft.  =  28.32  liters.    Ap.  %  X 100 

"      =7.481  gal.  (U.S.).   Ap.  a% 

1  bushel  (U.S.)  =  1.244  cb.ft.  Ap.  1M 

"   =0.352  4  hectoliter.  Ap.%-J-10 

1  hectoliter  =  26. 42  gals.  (U.S.)  Ap. 

%X10 


1  cb.  yd.  =  0.7646  cb. meter.    Ap.  ^ 
1  cb.  m.  =  35.31  cb.  ft.     Ap.  7/2X10 

=  1.308  cb.  yds.    Ap.  1M 
Digit  conversion  tables,  p.  51 

Weights:  complete  table,  p.  57-61 
1  grain  =64.80  mg.    Ap.  65 
1  gram  =  15. 43  grains.     Ap.  15^ 

=  0.035  27  oz.  Ap.%-^-100 
1  ounce  =28.35  grams.  Ap.  zh  X 100 
1  pound  (av.)  =0.453  6  klg.  Ap.  %o 
jl  kilogram  =  35  27  oz.  Ap.  %  X  10 
=  2.205  pounds.  Ap.  2M> 
t  short  ton  =0.907  2  met.  ton.  Ap. 

subtr.  Vi0 

lo.  =  0.892  9  long  ton.   Ap.  subtr.  Mo 
metric     ton  =  1.102    short    tons. 

Ap.  add  Vio 

lo.  =0.984  2  long  ton.     Ap.  1 

long  ton  =  1.12  short  tons.  Ap.  *% 

=  1.016  met.  tons.     Ap.  1 

3igit  conversion  tables,  p.  59 

(£  Weights  and   Lengths;   Wt.   of 

Bars:    complete  table,  p.  62-63 

Ib.  per  mile  =  0.281  8  kg.  per  km. 

Ap.  %  m 
•I  kg.  per  kilometer  =  3. 548  Ibs.  per 

mile.     Ap.  % 
1    Ib.    per   yard  =  0.496  1    kg.    per 

meter.     Ap.  H 
1    kg.    per    meter  =  2. 016    Ibs.    per 

yard.     Ap.  2 

do.  =0.672  0  Ib.  per  ft.     Ap.  ?£ 
llb.perft.  =  1.488  kg. perm.   Ap.  % 
Pressures:  complete  table,  p.  64-67 
1  Ib.  per  sq.  ft.  =  4. 882  kg.  per  sq. 

meter.     Ap.  4% 
1  ft.  water  column  =  62. 43  Ibs.  per 

sq.  foot.     Ap.  60% 
do.  =0.029  50  atm.     Ap.  8/iro 
1  Ib.  persq.  in.  =0.070  31  kg. per  sq. 

cm.    Ap.  "vioo 

do.= 0.068  04  atmosphere.     Ap.%0 
1    kg.   per   sq.  cm.  =  14. 22   Ibs.  per 

sq.  in.    Ap.  10% 
do.  =0.967  8      atmosphere.         Ap. 

subtr.  Vao 

1  atm.  =  14.70  Ibs.  per  sq.  in.  Ap.  4% 
"     =1.033  kg.  per  sq.  cm.     Ap. 

add  Vac 

Digit  conversion  tables,  p.  67 

Weights  and  Volumes:  complete 
table,  p.  68-69 

1   Ib.  per  cb.  yd.  =0.593  3  kg.  per 

cubic  meter.    Ap.  %o 
kg.  per  cb.  meter=  1.686  Ibs.  per 

cb.  yd.    Ap.  10/6 
do.  =0.062  43  Ib.  per  cb.  ft.    Ap.  %0 
,1  Ib.  per  cb.  ft.  =  16. 02  kg.  per  cb. 

meter.     Ap.  8% 

Weights     of     Water:     complete 
table,  p.  70 

1  cb.  cm.  =0.035  27  oz.    Ap.  %oo 
1  cb.  inch  =16. 39  grams.    Ap.  10% 
=0.578  0  ounce.    Ap.  H 
1  pint  =  1.043  pounds.     Ap.  add  ^20 


CONDENSED   LIST. 


1  liter  =  2. 205  pounds.    Ap.  22/i0 
.  Icb.  ft.  =  62.43  pounds.   Ap.%XlOQ 
"      =28.32  kilograms.    Ap.  20% 
Icb.  yd.=1686.1bs.   Ap  MX  10000 
=  764.6  kg.    Ap.  MX  1  000 
=  0.752  5  ton  (long).   Ap.M 
lcb.meter  =  2  205.  pounds 

=  0.984  2  ton  ( long).  Ap  1 
Volumes    of     Water:     complete 
table,  p.  71 

1  gram  =  0.061  02  cb.  in.     Ap.  9ioo 
1  ounce  =  28.35  cb.  cm.    Ap.  2/7  X  100 
1  Ib.  =  27.68  cb.  in.    Ap.  1J^iX10 
"  =0.453  6  liter.    Ap.  Vn 
"  =0.016  02  cb.  foot.  Ap.%-^100 
1  kilogram  =61.02  cb.  in.    Ap.  60 

"    =0.035  31  cb.ft.  Ap.y2-3- 100 
Energy;  Work;   Heat:    complete 
table,  p.  74-77 

1  joule  =  0.737  6  ft.-lb.    Ap.  M 
=  0.238  9  small  calorie. 
=  0.1020  kg. -met.    Ap.  Ho 
1  ft.-lb.  =  1.356  joules.     Ap.  ^ 

=  0.3239  small  cal.     Ap.  i%o 
=  0.138  3  kg. -met.    Ap.  %o 
"       =0.001  285  thermal  unit.  Ap. 

% -5- 1000 

1  kg.-m.=9.806   joules.    Ap.    10 
=  7.233  ft.-lbs.    Ap.  so/ii 
=  2.342  g. -calories.  Ap.  % 
1  ther.  unit  =  l  055.  joules.  Ap.  210% 
"     =778.1  ft.-lb.    Ap.  700% 
4     =107.6kg.-met.  Ap.  108 
"     =0.252  0  kg.-cal.    Ap.  \i 
1  watt-hour  =  2  655. ft.-lbs.  Ap.£00% 
=  367.1  kg.-met.  Ap.110% 
=  0.860  Okg. -gal.  Ap.% 
1  cal.  (kg.)  =  4  186.  joules.    Ap.  4  200 
=  3088.  ft.-lbs.   Ap.  3  100 
=  426.9  kg.-met.   Ap.300^ 
=  3. 968  ther.  units.  Ap.4 
=  1.163  watt-hrs.    Ap.  % 
1  met.  hp.-hr.  =  l  952  910.  ft.-lbs. 
=  270000.  kg.-met. 
1  hp.-hr.  =  l  980  000.  foot-pounds 

=  273  7 45. kilogram-meters 
1  kw.-hr.  =  2  655  403.  foot-pounds 

'     =367  123.  kilogram-meters 
Digit  conversion  tables,  p.  77 

Relations  between  Torque  and 
Energy,  p.  78 

Traction  Energy,  p.  78 

1  ton  (met.)-km.  =0.684  9  ton  (sh.)- 

mile.    Ap.  9/i3 
1  ton  ( sh.)-mile  =  1.460  ton  (met. )- 

kilometer.    Ap.  l% 
Tractive  Force,  p.  78 

1  Ib.  per  ton  (sh.)  =0.500  0  kilogram 

per  ton  (met.) 
1  kg.  per  ton  (met.)  =  2. 000  pounds 

per  ton  (short) 

Power:     complete  table,  p.  80-82 
1  watt  =44.26    ft.-lbs.   per  minute. 

Ap.  %X100 
do.  =  14. 33   gram -cal.    per    minute. 

Ap.  Vr  X  100 

do.  =6.119    kg.-met.    per    minute. 
Ap.  6 


1  met .  hp .  =  7  5 . 00     kilogram  -m  eters 

per  sec.    Or  MX  100 
do.  =0.986  3  hp.    Ap.  1 
do.=0.7354  kw.     Ap.22/3-^10 
1  hp.  =  33  000.    ft.-lbs.     per     min. 

Ap.  MX  100  000 
"    =1.014  metric  hp.    Ap.  1 
"    =0.7457  kilowatt.    Ap.  M 
1  kw.  =  1.360  m.  hp.    Ap.  add  $i 

4     =1.341  hp.    Ap.  add  H 
Digit  conversion  tables,  p.  82 

Forces:  complete  table,  p.  83 

1  dyne  =  1 .020  milligrams.    Ap.  1 
1  gram  =  980. 6  dynes.    Ap.  1000 
1  pound  =444  791.  dynes 
Moments     of    Inertia    and    of 

Momentum,  p.  84 

Linear       Velocities:       complete 

table,  p.  85-86 

1  km.  per  hr.'=16.67  met.  per  min. 

Ap.  MX  100 
1  ft.  per  sec.  =0.681  8  mile  per  hour. 

Ap.  68 -MOO 
1  mile  per  hour  =  1 .467  ft.  per  second. 

Ap.  1% 

1  meter  per  sec.  =  3. 6  km.  per  hour 
Angular  Velocities,  p.  86 

Frequency,  p.  86 

Linear  Accelerations:    complete 

table,  p.  88 

Gravity  =  32.17  ft.  per  sec.  per  sec. 

Ap.  32 

Gravity  ==9.806  m.  per  s.  per  s.  Ap.10 
Angular  Accelerations,  p.  88 
Angles:  complete  table,  p.  89 

1  deg.  =0.017  45  radian.  Ap.  %-*- 100 
1  radian  =  57 .30  degrees 
1  right  angle  =  1.571  radians.  Ap.  ^ 
Solid  angles,  p.  89 

Grades:  complete  table,     p.  90-91 
1  foot  per  mile  =  0.018  94% 
1%  =52.80  feet  per  mile 
Conversion  table    1  to  15%,     p.  92 
Time,  p.  94 

Discharges;     Flow    of    Water; 

Irrigation  Units,  p.  95 

1  miner's  inch  =  1.5  cb.  ft.  per  min. 
1  acre-foot  =  325  851.  gallons 
Electric  and  Magnetic  Units: 
Mean,  effec.  and  max.  values,   p.  97 
Resistance,  Impedance,  React- 
ance, p.  99 
1    Brit.  Assn.  unit  =0.986    7    ohm. 

Ap.  subtr.  1% 
lohm=  1.014  Brit.  Assn.  units.     Ap. 

add  1% 

Resistance  and  Length,     p.  101 

Resistance  and  Cross-section, 

p.  10) 

Resistivity;      Specific      Resist 

aiice,  complete  table,  p.  103-4 
1  ohm,  circ.  mil,  ft.,  unit  =0.001  662 

ohm,  sq.mm.,m.,unit.  Ap.  %oo 
1   ohm,   sq.   mm.,  m.,   unit  =  601. 5 

ohm',  circ.  mil,  foot  units 
Resistivity  of  pure  copper: 
=  10.03  ohm,  circ.  mil,  foot  units 
=  0.016  67  ohm,sq.mm.,m.,unit 


CONDENSED    LIST. 


Resistivity  of  mercury: 
=  565.9  ohm,  circ.  mil,  foot  units 
=  0.940  7  ohm,  sq.  mm.,  meter  unit 
Conductance,  Admittance,  Sus- 
ceptance,  p. 105 

Conductivity,  Specific  Conduc- 
tance, complete  table,    p.  107 
1  mercury  unit=0.017  72 copperunit 
1  copper  unit  =  55. 7 6  mercury  units 
Electromotive   Force,  complete 
table,  p.  109-111 

1  volt  =  0.981  0  Weston  cell  at  20°  C. 

Ap.  subtr.  2% 

"     =  0.697  4  Clark  cell  15°.  Ap.  %o 
1  Weston  cell  at  20°  C.: 
=  1.019  4  volts     Ap.  add  2% 
=  0.710  9  Clark  cell  at  15°  C.    Ap.  % 
1  Clark  cell  15°  =  1.434  volts.  Ap.  i<ft 
=  1 .407  Weston  cells 
at  20°.    Ap.  % 

E.M.F.  of  Clark  and  Weston  cells  at 

different  temperatures,     p.  Ill 

Current,  p.  113 

Current  Density,  complete  table, 

p. 114-115 

1  ampere  per  square  inch: 
=  0.155  0  amp.  per  sq.  cm.  Ap.  -/\% 
1  ampere  per  square  centimeter:     . 
=  6.452  amp.  per  sq.  in.    Ap.  6^ 
1  ampere  per  square  millimeter: 
=  0.0005067    ampere   per  circular 

mil.     Ap.  ^-s-1000 
1  ampere  per  circular  mil: 
=  1974.  amp.  per  sq.  mm.  Ap.2000 
Quantity;  Charge,  p.  116 

Capacity,  p.  117 

Inductance,  p.  119 

Time  Constant,  p.  120 

Frequency:  complete  table,  p.  121 
Frequency  =  0.159  2  X  angular  veloc. 
Angular  velocity  =  6.283  X  frequency 
Kinetic  Energy  of  a  Current, 
p.  122 
Electrical  Energy,  p.  123 

(see  also  under  Energy,  p.  74) 
Electrical  Power,  p.  125 

(see  also  under  Power,  p.  80) 
Electrochemical  Equiv.,  p.  126 
Ionic  charge  =  96  539,  coulombs 
Electrolytic  Deposits,  p.  126-7 
1  pound  per  day  =  0. 182  6  ton  (short) 

per  year.     Ap.  *%-*-10 
1  kg.  per  day  =  0.365  3  met.  ton  per 

year.     Ap.*%*t-10 
Electrochemical  Energy:  com- 
plete table,  p.  129 
Volts  =  kg. -calories  per   gram-mole- 
cule X  0.043  36.    Ap.^o 
Magnetic  Reluctance,         p.  129 
Oersteds  =  gilberts  -*-  maxwells 

=  ampere- turns  X  1.257  -*- 
maxwells 

Magnetic  Reluctivity,         p.  130 
Magnetic  Permeance,         p.  131 


Magnetic  Permeability,    p.  132 

=  gausses  in  iron  -=-  gausses  in  air 
Magnetic  Susceptibility,  p.  132 
Magnetomotive  Force:  complete 

table,  p.  133-4 

1    gilbert  =  0.795  8    ampere-turns. 

Ap.  subtr.  ^ 

1  amp.-turn  =  1.257gilb't.  Ap.add^ 
Gilberts  =  maxwells  X  oersteds 
Magnetizing  Force,          p.  136-7 
1  amp.  -turn  per  inch  =  0.494  7  gil- 

bert  per  centimeter.       Ap.  y% 
1  gilbert  per  cm.  =2.021  amp.  -turn 

per  inch.     Ap.  2 

Magnetic  Flux,  p.  138-9 

Maxwells  =  gausses  Xsq.  centimeters 

=  gilberts  •*-  oersteds 
Magnetic    Flux    Density:    com- 

plete table,  p.  141-2 

1  inch-gauss  =  0.1550  gauss.  Ap.%3 
1  gauss  =  6.452  inch-gausses.  Ap.  6^2 
Gausses  =  maxwells-:-  sq.  centimeters 
Magnetic  Moment,  p.  142 

Intensity  of  Magnetization,  142 
Magnetic  Energy,  p.  143 

Magnetic  Power,  p.  144 

Photometric  Units: 
Intensity     of    Light;      Candle 

Power:  complete  table,  p.  146 
1  hefner  =  0.88  English  candle 
1  English  candle  =  1.14  hefners 
Flux    of    Light;     Spherical    or 
Hemisph.  Candle  Power,  147 
Illumination:  compl.  table,  p.  148 
1  met.-cp.  =  0.081  8  ft.-cp.     Ap.  ^2 
1  ft.-cp.  =  12.  2  met.-cp  (hefners) 
Brightness  of  Source,  p.  148 

Quantity  of  Light,  p.  149 

Light  Efficiency,  p.  149 

Thermometer  Scales: 
Reduction  factors,  p.  150 

Readings  in: 
C.°  =  (F.°-32)X% 


.  . 

Scales:       Centigrade,      Fahrenheit] 

Reaumur,  Absolute   and   Con- 

crete,  from    absolute    zero    to 

6000°  C.,  p.  151-163 

Money,  Foreign,  p.  164 

Money  and  Length,  p.  166 

Money  and  Weight,  p.  167 

Scales  of  Maps  and  Drawings, 

p.  168 

Functions  of  IT,  p.  169-170 

Useful  IS  umbers,  p.  170 

Systems  of  Logarithm  g,     p.  171 
Acceleration  of  Gravity,    p.  171 
Mechanical        Equivalent        of 
Heat,  p.  171 

Specific  Heat  of  Water,      p.  171 
Miscellaneous    Foreign    Meas- 
ures, p.  172 

Index,  p.  175-196 


READY-REFERENCE  TABLES 


FROM  -THE-  LI  BRARY-  OF 
•WILLIAM -A  HILLEBRAND 


ENGINEERING  LIBRARY 


READY 
REFERENCE  TABLES. 


CONVERSION  FACTORS 

OF  EVERY  UNIT  OR  MEASURE  IN  USE,  INCLUDING  THOSE  OF 

LENGTH,     SURFACE,     VOLUME,      CAPACITY,      WEIGHT,     WEIGHT     AND     LENGTH, 
PRESSURE,    WEIGHT    AND    VOLUME,    WEIGHT    OF    WATER,    ENERGY,    HEAT, 
POWER,      FORCE,      INERTIA,      MOMENTS,      VELOCITY,      ACCELERATION, 
ANGLES,      GRADES,      TIME,       ELECTRICITY,       MAGNETISM,      ELEC- 
TROCHEMISTRY,     LIGHT,      TEMPERATURE,      MONEY,      MONEY 
AND    LENGTH,    MONEY    AND    WEIGHT,    NUMEROUS    COM- 
POUND UNITS,  USEFUL  FUNCTIONS  AND  NUMBERS, 
ETC.,      ETC.       WITH     THEIR      ACCURATE      AND 
THEIR     APPROXIMATE     VALUES,     THEIR/ 
LOGARITHMS,       RELATIONS,      DIGIT 
CONVERSION  TABLES,    EXPLA- 
NATIONS    O  F     CALCULA- 
TIONS,    ETC.,     ETC. 


BASED  ON  THE  ACCURATE  LEGAlllSl 

JNITED  STWES. 


Past  1 


LARD  VALUES  OF  THE 


CONVEN 
ENGINEERS,  PHY! 


/ARRANGED  FOR 
§7  STUDENTS,  MERCHANTS,  ETC. 


BY 


CARL  HERING,  M.E., 

^esident  American  Institute  of  Electrical  Engineers; 
President    Engineers'   Club  of  Philadelphia; 
Delegate  to  International  Congresses ;  etc. 


F*  I  Fl  3  '??.    Kf  D  I  T  JC  Q  N  • 
TOTAL   ISSUE   EIGHT  THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,   INC. 
LONDON:  CHAPMAN  &  HALL,  LIMITED 


ENGINEERING  L1BRARV 


Copyright,  1904. 

BY 

CARL  HERING. 
Entered  at  Stationers'  Hall. 


PRESS  Of 

BRAUNWOHTH    &   CO. 

BOOK  MANUFACTURERS 

BROOKLYN,   N.  Y. 


*'/  look  upon  our  English  system  as  a  wickedly  brain-destroying  piece  of 
bondage  under  which  we  suffer.  I  say  this  seriously.  I  do  not  think  any 
one  knows  how  seriously  I  speak  of  it." — Sir  WILLIA.M  THOMSON  (now  Lord 
Kelvin),  Philadelphia,  Sept.  24,  1884. 


PREFACE. 

THE  present  is  the  first  of  several  volumes  in  preparation  by  the  author, 
which  are  intended  to  contain  collections  of  data  conveniently  arranged 
for  ready  reference.  In  this  first  volume,  all  the  various  measures  used 
in  practice,  more  especially  by  engineers  and  physicists,  are  given  with 
their  values  in  terms  of  as  many  of  the  others  as  they  are  likely  to  be  con- 
verted into  in  practice;  the  reciprocals  of  these  are  also  given,  thus  enabling 
every  calculation  involving  the  conversion  of  one  measure  into  another 
to  be  reduced  to  a  single,  simple  multiplication.  Moreover,  they  are  stated 
in  such  a  form  that  errors  due  to  dividing  instead  of  multiplying  are  entirely 
avoided.  It  has  been  the  intention  to  include  every  unit  or  measure  used 
in  practice,  besides  many  that  are  now  obsolete  but  are  occasionally  met 
with.  The  more  usual  foreign  units  or  measures  have  also  been  added. 

These  conversion  factors  are  not  compiled,  but  have  all  been  especially 
recalculated  for  this  volume,  •  under  the  direction  of  the  author,  from  the 
exact  legal  values  as  far  as  such  values  existed,  and  from  the  very  best, 
most  standard,  and  most  authoritative  values  obtainable,  when  no  legal 
values  existed.  The  greatest  possible  care  was  taken  in  selecting  these 
fundamental  values.  They  were  obtained  from  the  National  Bureaft  of 
Standards,  the  Director  of  the  Nautical  Almanac,  the  International  Geo- 
detic Association,  the  U.  S.  Coast  and  Geodetic  Survey,  the  U.  S  Treasury 
Department,  the  adoptions  and  recommendations  of  international  con- 
gresses and  national  societies,  standard  works  of  reference,  and  personal 
authorities,  preference  in  the  few  cases  of  disagreement  being  generally 
given  in  the  order  named.  The  authority  for  each  fundamental  value  is 
given.  As  all  the  conversion  factors  were  calculated  from  the  same  set  of 
fundamental  values,  they  are  all  consistent  with  each  other,  forming  a 
single,  interconvertible,  uniform  system.  It  is  believed  that  this  is  the 
first  time  such  a  complete  set  of  conversion  factors  of  all  measures  has  been 
published,  based  on  the  accurate  legal  standard  values.  It  is  also  believed 
to  be  the  first  complete  collection  of  all  the  electric,  magnetic,  and  photo- 
metric units,  together  with  their  interrelations,  that  has  been  published 
in  convenient  tabular  form  for  ready  reference. 

Accuracy,  completeness,  and  convenience  for  ready  reference  were  the 
chief  aims  in  the  preparation  of  these  tables.  The  calculations  were  very 
carefully  made,  in  many  cases  by  two  entirely  different  methods,  and  the 
resulting  values  were  checked  and  cross-checked,  often  several  times.  For 
most  of  the  values  a  final  comparison  was  made  between  the  electrotyped 
plates  and  the  original  calculation  sheets.  All  values  marked  with  aster- 
isks were  checked  by  the  original  authorities  after  the  pages  were  electro- 
typed.  It  is  therefore  believed  that  there  are  no  errors,  but  should  any 
errors  or  omissions  be  found,  the  author  would  greatly  appreciate  being 
notified  of  them,  as  it  is  the  intention  to  have  the  values  correct  and  the 
collection  complete,  so  as  to  serve  as  a  reliable  standard  of  reference. 

A  new  feature  has  been  added  in  the  form  of  convenient  approximate 
values  consisting  often  of  only  one  and  generally  of  only  two  digits.  These 
digits  have  been  so  chosen  that  they  reduce  the  calculation  to  the  simplest 
possible;  they  will  be  found  to  suffice  for  most  of  the  ordinary  computa- 
tions, being  always  correct  within  2%.  The  correct  logarithms  have  also 
been  given  for  nearly  all  of  the  conversion  factors. 

Another  new  feature  not  generally  found  in  tables  of  conversion  factors 
is  that  the  values  of  most  of  the  compound  units  are  given.  This  saves 
double,  triple,  or  even  more  lengthy  calculations,  in  which  errors  are  likely 
to  be  made,  as  they  often  involve  both  multiplications  and  divisions. 

A  table  of  physical  quantities  has  been  added  giving  the  derivation, 
symbols,  dimensional  formula,  etc.,  of  each;  the  table  is  similar  to  the 
one  approved  by  the  International  Congress  at  Chicago,  but  includes  many 


995716 


X  PREFACE. 

The  usual  method  of  giving  tables  of  "Weights  and  Measures"  has  been 
entirely  abandoned  here  as  too  cumbersome,  inadequate,  and  entirely 
impracticable  for  giving  numerous  values  each  with  its  reciprocals;  sucn 
tables  are  moreover  quite  unsuited  for  ready  reference  The  author  has, 
instead,  adopted  a  system  in  which  all  interconvertible  units  (like  the 
mechanical,  neat,  and  electrical  units  of  energy,  for  instance)  are  given 
together  in  one  table,  and  are  there  placed  in  the  order  of  their  size.  This 
system  is  not  only  a  more  condensed  and  practical  one,  but  is  also  far  more 
convenient  for  ready  reference. 

The  present  tables  are  an  extension  and  entire  recalculation  of  the  very 
much  smaller  ones  published  by  the  author  about  twenty  years  ago  under 
the  title  of  "Equivalents  of  Units  of  Measurement." 

Unusual,  foreign,  and  obsolete  units,  or  those  used  for  special  trades, 
have  been  added  as  far  as  they  were  obtainable,  and  in  every  case  their 
values  are  given  in  terms  of  the  usual,  legal,  or  modern  units. 

For  some  units,  more  particularly  the  electric,  magnetic,  photometric, 
etc.,  explanations  have  been  added  of  the  meaning,  derivation,  and  proper 
use  of  the  units,  which  are  often  incorrectly  understood  or  applied.  Some 
explanatory  notes  on  the  usual  methods  and  accuracies  of  calculations 
are  also  given  in  the  introduction. 

The  index  has  been  made  very  complete,  and  in  addition  to  this,  atten- 
tion is  called  to  the  Condensed  List  beginning  on  the  inside  cover-page, 
which  contains  the  most  frequently  used  values  with  page  references  to 
the  others. 

A  comparison  with  other  tables  will  show  that  many  of  the  latter  are 
not  based  on  the  legal  standards  of  the  country.  It  is  not  generally  known, 
ior  instance,  that  the  legal  foot  is  derived  from  the  meter  and  not  from  a 
standard  yard.  Many  of  these  other  tables  are  based  on  an  unauthorized 
vpJue  of  the  meter  in  inches. 

Although  the  accuracy  adopted  throughout,  namely,  six  places  of  figures 
and  seven  of  logarithms,  may  not  always  be  warranted  by  the  accuracy  of 
some  of  the  fundamental  figures,  yet  it  has  been  maintained  uniform 
throughout  these  tables  in  order  to  enable  changes  to  be  made  by  mere 
proportion,  when  more  accurate  fundamental  values  become  obtainable  in 
the  future. 

The  author  takes  pleasure  in.  expressing  his  appreciation  of  the  valuable 
assistance  contributed  by  others,  which  has  added  greatly  to  the  com- 
pleteness and  reliability  of  the  information  and  makes  mucn  of  it  authori- 
tative. He  desires  to  express  his  obligations  to  the  following:  To  the 
National  Bureau  of  Standards,  and  in  particular  to  its  Director,  Prof  S.  W. 
Stratton,  for  very  full  and  complete  published  and  unpublished  data  con- 
cerning the  values  of  the  legal  standards  of  this  and  other  countries;  also 
for  his  consent  to  allow  the  name  of  that  Bureau  to  be  added  to  these  data, 
and  for  much  other  valuable  information  and  assistance.  To  L.  A.  Fischer, 
Assistant  Physicist  of  that  Bureau  for  the  laborious  work  of  checking  the 
lorrectness  of  many  values  of  the  conversion  factors  of  the  fundamental 
mechanical  units.  To  Dr.  F.  A.  Wolff,  Assistant  Physicist  of  that  Bureau, 
for  similar  service  in  connection  with  the  standard  values  of  some  of  the 
electrical  units.  To  Prof.  Walter  S.  Harshman,  Director  of  the  Nautical 
Almanac,  for  his  revision  of  the  table  of  units  of  time.  To  Prof.  Thomas 
Gray,  author  of  the  Smithsonian  Physical  Tables,  for  suggestions  and 
recommendations  concerning  some  of  the  derivational  and  dimensional 
formulas  in  the  table  of  Physical  Quantities.  To  Prof.  W.  S.  Franklin 
for  recommendations,  suggestions,  and  a  revision  of  those  relations  between 
the  electrical  and  magnetic  units  which  apply  to  varying  currents.  To  Dr. 
Ed.  L.  Nichols  for  information  concerning  some  of  the  standards  of  candle- 
power.  To  Dr.  A.  E.  Kennelly  for  a  revision  of  parts  of  the  table  of  physi- 
cal quantities.  To  the  author's  former  assistant,  Dr.  E.  F.  Roeber,  for  his 
able  work  in  mathematical  physics  and  his  reliable  calculations  of  many  of 
the  values  in  the  tables. 

CARL  HERING. 

PHILADELPHIA,  April,  1904. 


TABLE  OF  CONTENTS. 


Condensed  List inside  cover  page 

fymbols  and  Abbreviations xv-xv-'ii 

ntroduction ... 1-26 

Inter-relation  of  Units. .  . 1-4 

Compound  Names  of  Units 4 

Distinction  between  Units  and  Quantities  Measured  in  them.  .  4-5 

Reducing  Formulas  from  one  Kind  of  Units  to  Another 6-7 

Ratios 7 

Percentage 7-8 

Condensed  Numbers.     The  Use  of  10" 8-9 

Prefixes  Used  in  the  Metric  System 9 

Accuracy  of  Approximate  or  Abbreviated  Numbers 9-10 

Accuracy  of  Logarithms   10 

Absolute  System  of  Units.     The  C.  G.  S.  System 11-12 

Dimensional  Formulas. 12-14 

Decisions  of    International  Electrical  Congresses   concerning 

Electric,  Magnetic,  and  Photometric  Units  and  Definitions.  14-16 

Tables  of  Physical  Quantities  and  Relations 17-26 

Fundamental 18 

Geometric > 18 

Mechanical , 18 

Magnetic,  electromagnetic  system 20 

1 '          electrostatic  system 21 

Electric,  electromagnetic  system 22-23 

"        electrostatic  system 24-25 

Electrochemical,  electromagnetic  system 23 

electrostatic  system 25 

Photometric 25 

Thermal 26 

TABLES  OF  CONVERSION  FACTORS 27-173 

General  Remarks 27-29 

Lengths 29-40 

Text 29 

Table  of  Usual  Measures 30-31 

Table  of  Unusual,  Special  Trade   or  Obsolete  Measures 31-32 

Table  of  Foreign  Measures 33-34 

Tables  of  Inches  in  Fractions,  Decimals,  Millimeters,  and  Feet.  35-38 

Digit  Conversion  Tables,  1  to  100 39-40 

Surfaces 41-44 

Text 41 

Table  of  Usual  Measures 41-43 

Table  of  Unusual,  Special  Trade,  or  Obsolete  Measures 43-44 

Table  of  Foreign  Measures 

Digit  Conversion  Tables 

Volumes ;  Cubic  and  Capacity  Measures 45-55 

Text 45-46 

Table  of  Usual  Measures 46-51 

Digit  Conversion  Tables 51 

Table  of  Unusual,  Special  Trade,  or  Obsolete  Measures 52-54 

Table  of  Foreign  Measures 64-55 

xi 


Xll  TABLE    OF  CONTENTS. 


PAGES 

Weights  or  Masses 56-61 

Text : 56-57 

Table  of  Usual  Measures 57-58 

Digit  Conversion  Tables 59 

Table  of  Unusual,  Special  Trade,  or  Obsolete  Measures 59-60 

Table  of  Relative  Weights  (used  in  Chemistry) 60 

Table  of  Foreign  Measures 60-61 

Weights  or  Masses  and  Lengths;    Weight  of  Wires,  Rails,  Bars; 

Forces, and  Lengths;    Film  or  Surface  Tension;    Capillarity.  .  62-63 
Pressures;  'Pressures  of  Water,  Mercury,  and  Atmosphere;   Stress 
or  Force  per  Unit  Area;    Weights  or  Forces  and  Surfaces; 

Weights  of  Sheets,  Deposits,  Coatings,  etc 63-67 

Digit  Conversion  Tables 67 

Weights  or  Masses  and  Volumes;    Densities;    Weights  of  Materi- 
als ;  Masses  per  Unit  of  Volume 67-69 

Weights  and  Volumes  of  Water 69-71 

Table  of  Weights  of  Water 70 

Table  of  Volumes  of  Water 71 

Energy;  Work;  Heat ;  Vis-Viva;  Torque 72-77 

Text 72-74 

Tables 74-77 

Digit  Conversion  Tables 77 

Relations  between  Torque  and  Energy 78 

Traction  Energy 78 

Tractive  Force;    Tractive  Effort;    Traction  Resistance;    Traction 

Coefficient .  .  .  .  :  .  . 78 

Power;  Rate  of  Energy ;  Rate  of  Doing  Work ;  Momentum 79-82 

Text 79 

Tables 80-82 

Digit  Conversion  Tables 82 

Forces ;  Weights  Considered  as  Forces 83 

Moments  of  Inertia.     Text .' 84 

Moments  of  Inertia  in  Terms  of  the  Mass 84 

Moments  of  Inertia  in  Terms  of  the  Surface 84 

Moments  of  Momentum ;   Angular  Momentum 84 

Linear  Velocities ;   Speeds 85-86 

Angular  Velocities,  Rotary  Speeds 86 

Frequency;   Periodicity;   Period;   Alternations. 86-87 

Linear  Accelerations;  Rate  of  Increase  in  Velocities;   Gravity.  .  .  .  87-88 

Angular  Accelerations;  Rate  of  Increase  in  Angular  Velocities.  ...  88 

Angles  (plane) ;   Circular  Measures 89 

Solid  Angles 89 

Grades;   Slopes;   Inclines 90-92 

Conversion  Table,  1  to  15% 92 

Time i 93-95 

Text 93 

Tabl-s 94-95 

Discharges;   Flow  of  Water;  Irrigation  Units;  Volume  and  Tune.  95 

Electric  and  Magnetic  Units 96-144 

General  Remarks 96 

Mean ,  Effective ,  and  Maximum  Values 97 

Resistance;   Impedance;   Reactance 97-100 

Resistance  and  Length  for  the  Same  Cross-section 101 

Resistance  and  Cross-section  for  the  Same  Length 101 

Resistivity;   Specific  Resistance 101-104 

Conductance;   Admittance;   Susceptance 104-105 

Conductivity;   Specific  Conductance ; 106-107 

Electromotive  Force;   Potential;   Difference  or  Fall  of  Poten- 
tial; Stress;  Electrical  Pressure;  Voltage 108-111 

E.  M.  F.  of   Clark  and  Weston  Cells  at   Different  Tempera- 
tures   Ill 

Electric  Current ;   Current  Strength  or  Intensity 112-114 

Current  Density 114-115 

Electrji  al  Quantity;   Charge 115-116 

Electrical  Capacity 117 

'uductance;  Coefficient  of  Self-  or  Mutual  Induction 118-120 


TABLE    OF    CONTENTS.  Xlll 


PAGES 

Time  Constant  (of  Inductive  Circuits) 120 

Frequency;  Periodicity;  Period;  Alternations 121 

Kinetic  Energy  of  a  Current  in  a  Circuit 122 

Electrical  Energy  or  Work 1 22-123 

Electrical  Power 124-125 

Electrochemical  Equivalents  and  Derivatives. 125-126 

Electrolytic  Deposits 126-127 

Electrochemical  EnePgy 128-129 

Magnetic  Reluctance;  Magnetic  Resistance 129-130 

Magnetic  Reluctivity:    Specific  Magnetic  Reluctance;    Mag- 
netic Resistivity;  Specific  Magnetic  Resistance 130 

Magnetic     Permeance;     Magnetic     Conductance;     Magnetic 

Capacity 130-131 

Magnetic  Permeability;    Specific  Permeance;    Magnetic  Con- 
ductivity   131-132 

Magnetic  Susceptibility , 132 

Magnetomotive   Force;    Ampere-turns;    Magnetic   Potential; 

Difference  of  Magnetic  Potential;  Magnetic  Pressure.  .  .  *  132-134 
Magnetizing   Force;    Magnetomotive   Force   per  Centimeter; 

Magnetic  Force;  Field  Intensity;  Magnetic  Calculations.  134-137 
Magnetic  Flux;    Lines  of  Force;    Flux  of  Force;   Amount  of 

Magnetic  Field;  Pole  Strength 137-139 

Magnetic  Flux  Density;   Magnetic  Induction;   Lines  of  Force 

per  Unit  Cross-section;  Earth's  Field 140-142 

Magnetic  Moment 142 

Intensity  of  Magnetization;   Moment  per  Unit  Volume;   Pole 

Strength  per  Unit  Cross-section 

Magnetic  Work  or  Energy 

Magnetic  Power ' 144 

Photometric  Units 144-149 

Intensity  of  Light;   Candle  Power.  .  . 144-146 

Flux  of  Light ;  Spherical  or  Hemispherical  Candle  Power 147 

Illumination 148 

Brightness  of  Source 

Quantity  of  Light 

Light  Efficiency;  Power  per  Candle  Power 149 

Thermometer  Scales 150-163 

Reduction  Factors  for  One  Degree 150 

Reduction  Factors  for  Readings  of  a  Temperature  in  Degrees.  150 

Tables  of  Values  from  Absolute  Zero  to  6000°  C .  151-163 

Money * 164-166 

Fluctuating  Currencies 166 

Money  and  Length 166 

Money  and  Weight 167 

Scales  of  Maps  and  Drawings 

Paper  Measure 

Miscellaneous  Measures 168 

Useful  Functions  of  TT 169-WO 

Useful  Numbers 170 

Systems  of  Logarithms 171 

Acceleration  of  G  ravity 171 

Mechanical  Equivalent  of  Heat.  . . 17 1 

Specific  Heat  of  Water 171 

Miscellaneous  Foreign  Measures 172-173 

Index 175 


SYMBOLS  AND  ABBEEVIATIONS 

USED  IN  THE  TABLES  AND  TEXT. 

The  page-numbers  refer  to  the  pages  where  the  explanation  or  most  impor- 
tant use  is  given. 


a     acceleration,  19 
A     acre,  43 

A,  a     ampere,  15,  113 
a     are,  43 

ah     ampere-hour,  116 

ap     apothecary  measures,  46,  57 

Aprx     approximate  within  2%,  28, 

96 

atm     atmosphere,  66 
av     avoirdupois  weights,  57 
a-t     ampere-turns,  132 
B     magnetic  induction,  140 

B,  b,  B,  b     susceptance,  22,  104 
B.A.U.     British    Association    unit, 

99 

bbl     barrel,  53 
Brit.     British  or  English,  30 
BTU     Board  of  Trade  unit  (kilo- 
watt-hour), 77 

BTU     British  thermal  unit,  74,  75 
bu     bushel,  50 
C,c,  C,c     capacity, electric,  23,  117 

C,  c     coulomb,  15,  116 

C°  Centigrade  degrees,  150 

c  cord,  54 

Cal     caloric,  large,  76 

cal     calorie,  small,  75 

Cal/min  calorie  (large)  per  min- 
ute, 81 

cal/min  calorie  (small)  per  min- 
ute, 80 

cal/s     calorie  (small)  per  second,  81 

cb     cubic.  46 

Ccm     circular  centimeter  42 

Cft     circular  foot,  42 

eg     centigram,  57 

C.G.S.  or  CGS  centimeter-gram- 
second,  11 

ch     chain,  32 

Gin     circular  inch,  42 

circ.     circular   41 

cl     centiliter,  52 

cm     centimeter,  30 

cm2     square  centimeter,  42 

cm3     cubic  centimeter,  46 

cm/s     centimeter  per  second,  85 


cm/s2     centimeter  per   second    per 

second,  88 

CM     circular  mil,  41 
Cmm     circular  millimeter,  41 
cp     candle  power,  144,  146 
cwt     hundredweight,  58 
d     diameter,  41 
dg     decigram,  57 
dkg     decagram  or  dekagram ,  59 
dkl     decaliter  or  dekaliter,  52 
dkm     decameter  or  dekameter,  32 
dks     decastere  or  dekastere,  54 
dl     deciliter,  47 
dm     decimeter,  30 
dm2     square  decimeter,  42 
dm3     cubic  decimeter,  48 
ds     decistere,  53 
dwt     pennyweight,  59 
dyne-cm     dyne-centimeter  (erg),  7 4 
dyne/cm     dyne  per  centimeter,  62 
dyne/cm2     dyne  per  square  centi- 
meter, 64 

e     base  of  Naperian  logarithms,  171 
e,  e     brightness  of  source  of  light, 

16,  148 
E,  e,  E,  e    electromotive  force,  22, 

108 
Et     electromotive  force  at  t  degrees, 

111 
E,  e     ell,  32 

E,  E     illumination,  16,  148 
eff .     effective  value 

elmg     electromagnetic,  96 
elst     electrostatic.  96 
e.m.f.     electromotive  force,  108 
F°     Fahrenheit  degrees,  150 
F     farad,  15,  117 

F,  f    force,  18 

F     magnetomotive  force,  132 

fl     fluid  measures,  52 

ft     foot  or  feet,  30 

ft2     square  foot,  42 

ft3     cubic  foot,  49 

ft-gr     foot-grain,  74 

ft-gr/s     foot-grain  per  second,  80 

ft-lb    foot-pound,  74 


XVI 


SYMBOLS    AND    ABBREVIATIONS. 


ft-lb/min     foot-pound  per  minute, 

80 

ft-lb/s     foot-pound  per  second,  80 
ft/C     foot  per  hundred,  90 
ft /ft     foot  per  foot,  90 
ft/M     foot  per  thousand,  90 
ft/min     foot  per  minute,  85 
ft/ml     foot  per  mile,  90 
ft/s     foot  per  second,     85 
ft/s2     foot  per  second  per  second,  88 
fur     furlong,  32 

G,  g,  G,  g     conductance,  22,  104 
g     acceleration  of  gravity,  171 
g     gram,  58 
gal     gallon,  49 
gi     gill,  52 
gr     grain,  57 

g-C     gram -Centigrade  heat  unit,  75 
g-cm     gram -centimeter,  74 
g-cm/s     gram -centimeter    per    sec- 
ond, 80 

g/cm     gram  per  centimeter,  62 
g/cm2     gram  per  square  centimeter, 

64 
g/cm3     gram  per  cubic  centimeter, 

69 
g/dm2     gram  per  square  decimeter, 

64 

g/h     gram  per  hour,  127 
g/m     gram  per  meter,  62 
g/min     gram  per  minute,  127 
gr/in     grain  per  inch,  62 
gr/in3     grain  per  cubic  inch,  68 
H     heat,  26 
H     henry,  119 
H     hydrogen,  126 
H     magnetic  flux  density,  140 
H     magnetizing  force,  134 
h     hour,  duration,  94 
h     hour,  time  of  day,  93 
ha     hectare,  43 
hg     hectogram,  60 
Hg     mercury 
hhd     hogshead,  53 
hks     hektostere,  54 
hi     hectoliter,  50 
hp     horse-power,  81 
hp-h     horse-power  hour,  77 
hp-m     horse-power-minute,  76 
hp-s     horse-power-second,  7;5 
I,i,7,  i     current,  (elec.)  intensity  of, 

22, 112 

1,7     intensity  of  light,  16,  144 
I     intensity  of  magnetization,  142 
in     inch,  30 
in2     square  inch,  42 
in3     cubic  inch,  47 
in  Hg     inch  of  mercury  column,  65 
in/m      inch  per  mile,  90 
int.     international 
J,  j     joule,  15,74,  123,  143 
J     mechanical  equivalent  of  heat  ,26 
k     electric  inductive  capacity,   14, 

18,  23,  24 

k     dielectric  constant,  23 
K     moment  of  inertia,  19 
kg     kilogram,  58 
kg  cal     kilogram  calorie,  171 


kg-C     kilogram      Centigrade      heat 

unit,  76  , 

kg-km     kilogram -kilometer,  76 
kg-km/min     kilogram  -kilometer  per 

minute,  81 

kg-m     kilogram -meter,  75 
kg-m/min     kilogram -meter  per  min- 
ute, 80 
kg-m/s     kilogram -meter  per  second, 

81 

kg/cm2     kilogram  per  equare  centi- 
meter, 66 

kg/cm3     kilogram   per  cubic  centi- 
meter, 69 

kg/day     kilogram  per  day,  127 
kg/h     kilogram  per  hour,  127 
kg/hi     kilogram  per  hectf  liter,  68 
kg/km     kilogram  per  kilometer,  62 
kg/1     kilogram  per  liter,  69 
kg/m     ki'ogram  per  meter,  63 
kg/m2     kilogram  per  square  meter, 

64 

kg/m3     kilogram  per  cubic  meter,  68 
kg/mm2     kilogram  per  square  milli- 
meter, 66 

kg/t     kilogram  per  ton  (metric),  78 
kg/yr     kilogram  per  year,  126 
kl     kilpUter,  54 
km     kilometer,  30 
km2     square  kilometer.  43 
kw-h     kilowatt-hour,  77 
km/hr     kilometer  per  hour.  85 
km/hr/min     kilometer  per  hour  per 

minute,  88 
km/hr/s     ki'ometer    per   hour   per 

second,  88 

km/min     kilogram -meter  per  min- 
ute, 86 

kw     kilowatt,  82 
kw-m     kilowatt-minute,  76 
kw-s     kilowatt-second,  75 
L ,  1 ,  L ,  I     inductance ,  self-induction , 

23,  118 

L,  I     length,  18 
1     liter,  48 
Ib     pound,  58 
Ib-C     pound  Centigrade  heat  unit, 

75 
Ib-C/min     pound-Centigrade      heat 

unit  per  miute,  81 
Ib-F     pound-Fahrenheit  heat  unitt 

75 

Ib-F/min     do.  per  minute,  81 
Ib/bu     pound  per  bushel,  68 
Ib/day     pound  per  day,  127 
Ib/ft     pound  per  foot,  63 
lb/ft2     pound  per  square  foot,  64 
Ib/ft3     pound  per  cubic  foot,  68 
Ib/gal     pound  per  gallon,  68 
Ib/h     pound  per  hour,  127 
Ib/in     pound  per  inch,  63 
Ib/in2     pound  per  square  inch,  65 
Ib/in3     pound  per  cubic  inch,  69 
Ib/ml     pound  per  mile,  62 
Ib/qt     pound  per  qt,  68 
Ib/tn     pound  per  ton  (av),  78 
Ib/yd     pound  per  yard,  62 
lb/yd3     pound  per  cubic  yard,  68 


SYMBOLS    AND    ABBREVIATIONS. 


XV11 


Ib/yr    pound  per  year,  126 

li     link,  31 

log     logarithm,  171 

log  jo     common  logarithm,  171 

loKe     Naperian  logarithm.  171 

m     magnet  pole,  strength,  20 

M     mass,  18 

M     mechanical  equivalent  of  heat, 
171 

m     meter,  30 

m     minute,  duration,  94 

m     minute,  time  of  day,  93 

m2     square  meter,  43 

m3     cubic  meter.  50 

m/min     meter  per  minute,  85 

m/s     meter  per  second,  85 

m/sec2     meter  per  second  per  sec- 
ond, 88 

mg     milligram,  57 

mg/mm     milligram  per  millimeter, 
62 

mg/s     milligram  per  second,  127 

mil2     square  mil,  41 

min     minute,  94 

ml     milliliter,  46 

ml     mile,  31 

ml2     square  mile,  43 

ml-lb     m  ile-pound ,  7  6 

ml-lb/min     mib-pound  per  minute, 
81 

ml/hr     mile  per  hour,  85 

ml/hr/min     mile  per  hour  per  min- 
ute, 88 

ml/hr/sec     mile  per  hour  per  sec- 
ond, 88 

ml/min     mile  per  minute,  86 

mm     millimeter,  30 

mm2     square  millimeter,  41 

mm3     cubic  millimeter,  52 

mm  Hg     millimeter  of  mercury  col- 
umn  65 

mm/m     millimeter  per  meter,  90 

m.m.f.     magnetomotive  force,  132 

mo     month,  94 

moJ     gram  molecule,  60 

N     number  of  turns,  20 

n,  n     any  number 

n     frequency,  23,  87,  121 

na     nail,  31 

O     ohm,  15 

O     oxygen,  126 

oz     ounce,  58 

oz/h     ounce  per  hour,  127 

oz/min     ounce  per  minute,  127 

P     page 

p     pole  (length),  perch,  32 

p     pressure,  19 

P,  P  power,  activity  (mechanical, 
electric,  magnetic,  etc.),  19 
20,  23,  124,  144 

p.d.     difference  of  potential,  108 

"per"     between   two   units  means 
first  divided  by  second,  4 

phys.     physical  or  physics 

pk     peck,  49 

pt     pint,  47 

Q.  q.  Q>  a     quantity  of  electricity, 
22,  115 


Q,  Q     quantity  of  light,  16,  149 

qr     quarter,  31,  53,  60 

qt     quart,  48 

R,  r,  R,  r     resistance  (electric),  22, 

97 

R     Reaumur  degrees,  150 
R     reluctance,  129 
R     rood,  44 
r     rod,  32 
rev     revolution,  89 
rev/h     revolution  per  hour,  86 
rev/min     revolution  per  minute,  86 
rev/min/min     revolution  per  min- 
ute, per  minute,  88 
rev/min/s     revolution   per  minute 

per  second,  88 

rev/s     revolution  per  second,  86 
rev/s/s     revolution  per  second  per 

second,  88 

rhp     revolution  per  hour,  86 
rpm     revolution  per  minute,  86 
rps     revolution  per  second,  86 
s     second,  duration,  94 
•     second,  time  of  day,  93 
s     shilling,  165 
s     stere,  51 
S,  s     surface,  18 
sec     second 
sp     specific 

sp.  gr.     specific  gravity,  18 
sq     square,  41 
S.U.     Siemens  unit,  99 
subtr.     subtract 
T,  t    time,  18 
t     temperature  degrees,  111 
t     ton,  metric,  58 
t-km     ton-kilometer,  78 
t/cm2     ton  (metric)  per  square  cen- 
timeter, 67 
t/m2    ton      (metric)     per     square 

meter,  65 
t/m3    ton  (metric)  per  cubic  meter, 

69 

t/yr    ton  (metric)  per  year,  127 
tn     ton  (avoirdupois),  58 
tn-ml     ton-mile,  78 
tn/ft2     ton  (av)  per  square  foot,  66 
tn/in2     ton  (av)  per  square  inch,  66 
tn/yd3     ton  (av)  per  cubic  yard,  69 
tn/yr    ton  (ay)  per  year,  127 
U,  u,  U,  u     difference  of  potential, 

22, 108 

U.S.     United  States 
v     velocity,  19 

v,  v     velocity  of  light,  14,  21 ,  24,  96 
V,  V     volume,  18,  69 
V,  v     volt,  15,  110 
vol.     volume 
TV.w    watt,  15.  80,  125 
W     weights,  69 
W,  W     work,  energy    (mechanical, 

electric,  magnetic,  etc.), 

vis-viva,  impact,  19,  20, 

23,  122,  143 
w-h    watt-hour,  76 
w-s     watt-second,  74 
X,  x,  X,  x     reactance,  22,  97 
xe    capacity  reactance,  22 


XV111 


SYMBOLS   AND   ABBREVIATIONS. 


y 
Z 


Xp    magnetic  reactance,  22 

Y,  y,  F,  y     admittance,  22,  104 

yd     yard,  30 

yd2     square  yard,  42 

yd3    cubic  yard,  50 

r    year,  94 

,  z,  Z,  z,    impedance,  22,  97 


a.  (alpha)     angle,  18 

ft  (beta)     angle,  18 

r  (gamma)     conductivity  (electric), 

22,  105 

r     0.001  milligram,  59 
d  (delta)     density,  18 
0  (theta)     temperature,  13,  18,  26 
K  (kappa)     susceptibility,  20,  132 
fj.  (mu)     magnetic  permeability,  14, 

18,  20,  131 

n     micron  or  micro-meter,  31 
fifji     milli-micron,  31 
^  (nu)     reluctivity,  20,  130 
n  (pi)     angle  of  180°,  89 
n     ratio  of  circumference  to  diam- 

eter, 169 

P  (rho)     resistivity,  22,  101 
0  (phi)     angle  of  phase  difference, 

124 

0,  0  (phi)     magnetic  flux,  20,  137 
0     flux  of  light,  16,  147 
a)  (omega)     angular    velocity,    19, 
23,  86,  121 

(B     flux    density,   magnetic    induc- 

tion, 20,  140 
€     magnetic  capacity,  20 


JF     magnet9motive  force,  20,  132 

3C  magnetizing  force,  field  inten- 
sity, 20,  134 . 

3C     flux  density,  140 

3  intensity  of  magnetization,  20, 
142 

3TC    magnetic  moment,  20,  142 

(R     reluctance,  20,  129 

X     multiplication 

*  (hyphen)     between     two     units 

means  their  product,  4 
•*•     division,  first  quantity  divided 

by  the  second 

%     per  cent  or  per  hundred,  7 ,  90 
Voo      per  mil  or  per  thousand,  8,  90 
y     square  root 
$/     cube  root 
~"1     reciprocal 
J    square,  41 

*  square  root 

*  cube  or  cubic.  46 
»  cube  root 

10n     for  condensed  numbers,  8 
>     degree,  89 
'     foot 

'     minute,  89 
'     inch 
"     second,  89 
'"    line  (1/12  inch) 
£     pound  sterling,  165 
$     dollar  in  the  United  States,  166 
3    dram,  apothecary,  59 
^    ounce,  apothecary,  59 
9    scruple,  59 
CO   frequency,  87,  121 


INTRODUCTION. 


INTER-RELATION  OF  UNITS. 

In  order  to  establish  a  stable,  systematic  set  of  relations  between  the 
various  units  in  use,  the  first  requisite  is  to  find  which  the  proper  funda- 
mental relations  are,  otherwise  the  secondary  relations  may  have  different 
values  depending  upon  how  they  have  been  calculated,  that  is,  they  will 
form  an  unstable  system.  Such  simple  relations  as  those  between  pounds, 
ounces,  grams,  kilograms  etc.,  are  definitely  established  by  law  and  are 
well  known.  The  relations  between  units  of  length  and  units  of  capacity 
(volume)  are  slightly  less  simple  but  are  also,  or  at  least  should  be,  estab- 
lished by  law.  The  relation  between  what  are  commonly  called  weights 
(more  correctly  masses)  and  lengths,  becomes  somewhat  more  complicated, 
but  by  means  of  the  mass  of  water  these  two  kinds  of  units  become  defi- 
nitely connected  with  each  other,  at  least  in  the  metric  system,  and  from 
that  to  all  others;  this  relation  is  also  defined  by  law. 

But  with  the  more  widely  differing  units  the  relations  become  more 
complicated.  It  may  even  seem  at  first  thought  as  though  there  was  no 
relation  between  such  units  as  a  foot  and  a  horse-power  or  a  watt;  or 
between  a  pound  and  a  degree  of  a  thermometer,  or  between  a  foot  and  an 
ampere  of  electric  current,  etc.  Yet  all  such  units  can  be  shown  to  be  con- 
nected with  each  other,  often  more  simply  than  might  at  first  appear.  In 
a  set  of  values  of  different  units  in  terms  of  the  others  it  is  therefore  neces- 
sary to  find  and  establish  all  the  necessary  fundamental  relations,  and  no 
more,  or  else  the  system  is  not  a  stable  one,  and  the  derived  values  become 
different,  depending  upon  how  they  are  calculated. 

To  find  the  connecting  links  between  such  widely  different  units,  it  is 
necessary  to  reduce  them  to  some  unit  or  quantity  common  to  all,  as  a 
direct  numerical  equivalent  can  be  given  only  between  units  of  the  same 
kind.  This  common  unit,  or  quantity,  and  the  only  one,  is  energy.  Energy 
can  be  expressed  in  terms  of  combinations  of  all  the  different  units  com- 
monly used  in  practice,  and  by  expressing  the  same  amount  of  energy  in 
terms  of  these  different  combinations  of  units  their  relations  to  each  other 
may  be  established. 

An  analysis  shows  that  there  are  three  typically  different  sets  of  well- 
established  units  in  common  use,  in  terms  of  which  energy  is  expressed  or 
measured,  and  these  three  sets  or  groups  include  all  the  units  generally 
used. 

The  first  and  most  stable  or  fundamental  are  the  absolute  units  and  all 
those  having  a  known  and  invariable  relation  to  them;  these  include  the 
electrical  units,  which  are  the  newest,  and  were  wisely  based  on  the  absolute 
ones.  They  are  the  same  throughout  the  universe;  they  are  invariable 
and  are  independent  of  any  constant  of  nature,  except  perhaps  the  unit 
of  time,  which  depends  on  the  revolution  of  the  earth  around  the  sun,  but 
as  this  has  been  so  accurately  determined  it  may  safely  be  considered  here 
as  an  absolute  quantity;  it  is  at  least  invariable  throughout  the  universe. 

The  next  group  of  units  is  the  one  involving  the  constant  of  nature 
called  the  attraction  of  gravitation  of  our  earth,  and  they  are  therefore 
often  called  gravitation  units;  they  are.  therefore,  purely  terrestrial  units, 
and  would  be  quite  different  on  other  celestial  bodies,  and,  in  fact,  are, 
strictly  speaking,  different  even  on  different  parts  of  this  earth,  although 
for  most  purposes  they  may  be  considered  to  be  the  same.  They  include 
such  units  as  the  pound  or  kilogram  considered  as  weights,  the  horsa- 


2  INTRODUCTION. 

power  ay  vfsually  defined,  etc.  It  will  be  found  upon  investigation  that  this 
whole  group,  is  linked  "with  'the  first  group,  namely,  the  absolute  and  elec- 
trical units, 'through  the  value  01  the  acceleration  of  gravity,  a  terrestrial 
constant  of  nature,  and  unfortunately  one  which  is  variable  and  has  no 
universally  .accepted  normal  value.  If  some  standard  normal  value  of  this 
constant  ^vcre  c»n*yersaily  established  and  were  fixed  definitely,  this  group 
of 'units  would  be  connected  with  the  absolute  units  by  a  fixed  relation,  and 
the  former  would  then  all  have  definite  and  invariable  values  in  terms  of 
the  absolute  units.  The  acceleration  of  gravity  is  therefore  the  connect- 
ing link,  and  the  only  one,  between  these  two  groups.  Throughout  the 
tables  in  this  book  the  standard  value  of  the  acceleration  of  gravity  has 
been  taken  as  980.596  6  cm  (the  authority  is  given  under  units  of  Accel- 
eration). 

The  third  and  last  group  of  units  is  the  one  involving  a  property  of 
water,  which  is  also  a  constant  of  nature.  This  group  includes  such  units 
as  the  calorie,  the  thermal  unit,  the  degrees  of  thermometer  scales,  etc.  It 
will  be  found  upon  investigation  that  this  whole  group  of  units  is  linked 
with  the  first  group,  namely,  the  absolute  and  electrical  units,  through 
the  value  of  the  specific  heat  of  water;  and  moreover  this  is  the  only  con- 
necting link.  This  constant  differs  from  the  acceleration  of  gravity,  which 
is  the  connecting  link  between  the  first  and  second  groups,  in  that  it  has 
not  a  variable  value  like  gravity;  but,  on  the  other  hand,  its  value  has  not 
yet  been  determined  with  such  accuracy.  It  is  like  gravity  in  that  it  is  a 
constant  of  nature  whose  value  must  be  determined  by  experiment  and 
definitely  established  before  fixed  and  stable  relations  between  this  group 
of  units  and  the  absolute  units  can  be  established. 

The  relations  between  the  various  units  in  any  one  group  are,  as  a  rule, 
fixed  by  definition  or  by  law;  the  horse-power,  for  instance,  is  defined  in 
terms  of  the  foot-pound  and  the  minute;  or  the  calorie  is  defined  in  terms 
of  the  thermometer  scale  and  a  quantity  of  water.  Such  relations  are  there- 
fore established  and  require  no  experimentally  determined  constant  in 
order  to  express  their  relations  to  each  other;  no  matter  what  the  value 
of  the  specific  heat  of  water  is,  the  relation  between  the  calorie  and  the 
thermometer  scale  is  fixed.  But  when  we  wish  to  express  the  values  of 
any  of  the  units  of  one  group  in  terms  of  units  of  the  other,  then  the  numer- 
ical values  of  these  two  connecting  links,  namely,  the  acceleration  of  gravity 
and  the  specific  heat  of  water,  must  be  known.  If,  for  instance,  horse- 
powers are  to  be  expressed  in  absolute  or  electrical  units  like  watts,  the 
value  of  gravity  must  be  known ;  or  if  calories  are  to  be  expressed  in  abso- 
lute or  electrical  units  like  joules,  the  value  of  the  specific  heat  of  water 
must  be. known;  or  if  calories  are  to  be  expressed  in  foot-pounds,  then 
both  these  constants  of  nature  must  be  known. 

Moreover,  the  values  of  gravity  and  the  specific  heat  of  water  are  the  only 
two  constants  that  must  be  known  in  order  to  reduce  any  of  these  different 
units  of  energy  to  any  of  the  others.  By  accepting  or  fixing  a  definite 
value  for  each  of  these  two  constants  the  relations  between  all  the  units 
of  these  three  groups  become  linked  together  into  one  single  stable  system 
in  which  all  the  relations  between  any  of  the  units  remain  absolutely  the 
same  no  matter  how  they  have  been  calculated.  As  no  standard  values 
of  either  of  these  two  constants  have  as  yet  been  universally  accepted  or 
agreed  upon,  which  is  very  unfortunate,  the  writer  has,  in  the  preparation  of 
these  tables,  selected  certain  values  as  the  ones  which  seem  to  be  the  best 
that  exist  at  the  present  time,  and  which  at  least  have  a  semi-official  en- 
dorsement. 

As  the  establishment  of  fixed  values  for  these  two  constants  forms  an 
inflexible  stable  system  of  units,  there  must  exist  some  relation  between 
these  two  constants  themselves.  It  can  readily  be  shown  that  the  specific 
heat  of  water  divided  by  the  acceleration  of  gravity  gives  the  mechanical 
equivalent  of  heat,  when  all  are  reduced  to  the  same  terms.  The  value  used 
throughout  this  book  is  426.9  kilogram-meters  per  kilogram-Centigrade 
heat  unit  (see  authority  under  units  of  Energy).  When  thus  defined  this 
constant  becomes  a  secondary  or  derived  one.t  which  is  as  it  should  be, 

t  While  this  is,  strictly  speaking,  the  more  rational  way  of  deducing  those 
constants,  yet  the  author  has  preferred  to  accept  the  simpler  value  above 
given  for  the  mechanical  equivalent  and  has  made  the  specific  heat  the  in- 
commensurable derived  unit  for  reasons  explained  under  units  of  Energy. 


INTER-RELATION    OF  UNITS. 


because  it  does  not  involve  the  absolute  units,  but  is  the  relation  between 
the  second  and  third  groups,  both  of  which  are  separated  from  the  absolute 
units  by  one  of  these  two  empirical  and  therefore  less  stable  constants. 

If  the  heat  units  and  what  are  called  the  gravitational  units,  like  the 
pound  (weight),  the  horse-power,  etc.,  had  originally  been  denned  in  terms 
of  the  absolute  units,  as  was  wisely  done  with  the  electrical  units,  there 
would  have  been  no  necessity  of  knowing  the  values  of  gravity,  the  specific 
heat  of  water,  or  the  mechanical  equivalent  of  heat,  in  order  to  establish 
fixed  and  invariable  relations  between  all  the  units  in  use.  Such  fixed  rela- 
tions would  then  be  established  definitely  by  definition.  Concrete  stand- 
ards of  measurement  complying  as  closely  as  possible  with  the  defined 
values  would  then  require  the  experimental  determinations  of  these  con- 
stants, but  the  exact  theoretical  relations  would  not.  Such  is  the  case  with 
the  electrical  units  which  are  defined  with  absolute  accuracy  in  terms  of 
the  absolute  units  and  require  no  experimentally  determined  constant  to 
connect  them  with  the  absolute  units;  in  the  electrical  system  of  units  the 
problem  therefore  is  not  to  find  such  an  empirical  relation,  but  to  deter- 
mine concrete  standards  which  comply  with  the  defined  relations  as  closely 
as  possible. 

If  those  on  whom  the  duty  will  fall  to  establish  a  system  of  units  of  light 
would  follow  the  good  example  set  by  those  who  established  the  electrical 
units,  and  base  them  on  the  absolute  system  of  units,  instead  of  on  another 
different  constant  of  nature,  like  the  candle  or  incandescent  platinum,  they 
would  avoid  creating  a  fourth  group  of  units  whose  empirical  relation  to  the 
absolute,  electrical,  and  to  the  other  two  groups  of  units  would  have  to  be 
determined,  and  would  be  uncertain  until  it  was.  What  is  called  radiated 
light,  usually  measured  in  spherical  candle-power,  is  a  true  power  or  a  rate 
of  work,  and  the  absolute  unit  would  therefore  be  a  dyne-centimeter  per 
second.  If  the  practical  unit  could  be  defined  as  a  decimal  multiple  or 
fraction  of  this,  like  the  watt  is,  it  would  at  once  become  a  definitely  known 
unit.  It  would  then  become  necessary,  as  it  was  with  the  three  funda- 
mental electrical  units,  to  establish  a  concrete  standard  which  would  com- 
ply as  definitely  as  possible  with  the  defined  unit.  If,  however,  the  prac- 
tical unit  is  defined  on  some  independent  basis,  like  the  present  unsatis- 
factory and  uncertain  temporary  light  units,  it  will  become  necessary  to 
determine  experimentally  the  mechanical  equivalent  of  light,  before  any 
of  the  much-needed  relations  could  be  established  with  the  other  existing 
units,  like  the  electrical,  gravitational,  or  heat  units.  In  the  case  of  light 
units  the  matter  becomes  complicated  on  account  of  the  different  wave- 
lengths and  combinations  of  wave-lengths  which  have  different  effects  on 
the  eye  in  perceiving  light.  Owing  to  the  absence  of  any  known  relations 
between  light  energy  and  energy  stated  in  other  units,  no  relations  ^between 
those  groups  of  units  could  be  given  in  this  book. 

The  relations  above  described,  which  exist  between  the  different  groups 
of  units,  maybe  represented  graphically  as  shown  in  the  following  diagram: 


GRAVITATIONAL, 
UNITS  and  their 
derivatives;  gram 
or  pound  as  weights, 
foot-pounds,  horse- 
powers, etc. 


HEAT  UNITS  and 
their  derivatives; 
calorie,  thermal 
unit,  thermometer 
scale,  etc. 

LIGHT  UNITS  and 
their  derivatives; 
candle  power, 
lux,  etc. 


ABSOLUTE  or  ELECTRICAL  UNITS 
and  their  derivatives;  gram  or  pound 
as  masses,  dyne,  centimeter,  foot, 
second,  watts,  joules,  etc. 


4  INTRODUCTION. 

The  most  fundamental,  namely,  the  absolute  or  electrical  units,  are  shown 
at  the  bottom.  The  gravitational  units  to  the  left  are  .shown  linked  to 
these  through  the  value  of  the  acceleration  of  gravity.  The  heat  units 
to  the  right  are  shown  linked  to  the  absolute  and  electrical  units  through 
the  value  of  the  specific  heat  of  water.  Finally,  the  gravitational  units  are 
linked  to  the  heat  units  through  the  mechanical  equivalent  of  heat,  which 
is  the  quotient  of  the  other  two  linking  values.  The  relation  of  any  one 
of  the  three  to  either  one  of  the  others  can  be  found  either  by  means  of  the 
direct  link  or  indirectly  by  means  of  the  two  other  links;  the  result  must 
be  the  same  if  the  values  expressed  by  these  links  are  definitely  established. 

The  group  of  units  of  light  at  present  in  temporary  use  are  shown  as  a 
fourth  group,  the  connecting  links  of  which  are  unfortunately  not  yet  known, 
for  which  reason  no  relations  to  other  units  can  be  given,  although  such 
definite  relations  it  seems  ought  of  necessity  to  exist  at  least  when  light  is 
considered  as  a  radiation  of  some  definite  wave-length. 

COMPOUND   NAMES   OF  UNITS. 

When  in  compound  names  of  units  of  measurement  the  names  of  the  sim- 
ple units  are  joined  by  a  hyphen,  as  in  foot-pounds,  horsepower-hours,  etc.,  it 
signifies  the  product  of  the  simple  units;  that  is,  an  amount  of  energy  stated 
in  foot-pounds  means  that  the  number  of  feet  is  multiplied  by  the  number 
of  pounds,  to  give  the  foot-pounds;  in  other  words,  the  compound  quantity 
varies  directly  with  each  of  the  others.  If,  however,  the  simple  units  are 
joined  by  the  word  "per"  as  in  feet  per  second,  pounds  per  mile,  etc.,  it  means 
that  the  first  is  divided  by  the  second;  that  is,  a  velocity  in  feet  per  second 
means  that  the  number  of  feet  is  divided  by  the  number  of  seconds.  In 
other  words,  the  compound  quantity  varies  directly  with  the  first  and  in- 
versely with  the  second.  An  acceleration  in  "miles  per  second  per  second  " 
means  that  the  velocity  in  miles  per  second  is  divided  by  the  number  of 
seconds  during  which  this  velocity  was  acquired. 

The  above  is  the  correct  practice,  but  it  is  unfortunately  not  followed 
by  all  writers.  For  instance,  "horsepower  per  hour"  is  wrong;  it  should 
be  horsepower-hour,  as  the  number  of  hours  is  a  multiplier  and  not  a 
divisor. 

DISTINCTION   BETWEEN   UNITS   AND  QUANTITIES 
MEASURED   IN   THOSE   UNITS. 

Serious  errors  are  not  infrequently  made  by  failing  to  distinguish  between 
the  calculations  of  the  values  of  the  units  themselves  and  the  calculation  of 
quantities  measured  in  terms  of  those  units.  By  using  the  reduction  factors 
in  the  way  they  are  given  in  the  tables  in  this  book,  no  mistakes  can  be 
made  as  the  units  have  all  been  calculated,  and  the  value  of  each  unit  is 
given  in  terms  of  each  of  the  others;  they  are  just  like  the  selling  prices  of 
one  apple  in  terms  of  different  moneys,  the  only  remaining  calculation  is 
to  multiply  the  quantity  of  these  units  by  the  reduction  factor  given  in  the 
table;  thus,  if  1  foot-pound  =  0.14  kilogram-meter,  then  3  of  these  foot- 
pounds  =  3X0.14  kilogram-meters.  But  in  determining  the  reduction  fac- 
tors themselves,  or  in  similar  calculations,  which  are  not  infrequent,  errors 
are  very  apt  to  be  made  by  multiplying  where  one  should  divide.  A  fre- 
quent case  is  that  in  which  a  formula  reads  in,  say,  distances  in  meters  and 
weights  in  kilograms  and  it  is  desired  to  change  it  to  read  in  feet  and  pounds; 
this  will  be  described  in  the  next  section;  or,  in  dealing  with  compound 
units,  such  as  foot-pounds  (ft  X  Ibs)  and  feet  per  pound  (ft  •*-  Ibs),  which 
were  described  above. 

A  clear  distinction  should  be  made  between  the  units  themselves  and  the 
quantities  measured  or  expressed  in  terms  of  those  units.  Thus,  1  foot  =  12 
inches  is  the  value  of  one  unit  in  terms  of  another,  but  it  is  not  correct  to 
interpret  this  by  saying:  a  length  in  feet  =  12  X its  length  in  inches,  as  that 
would  be  144  times  too  great.  It  should  be  interpreted  as  follows:  If  1  foot 
equals  12  inches,  then  a  length  of  any  other  number  of  feet  must  be  mul- 
tiplied by  12  to  reduce  that  length  to  inches,  or  to  express  it  in  terms  of 
inches.  Simple  as  this  may  seem  in  this  almost  self-evident  illustration 
mistakes  are  easily  and  frequently  made  in  more  obscure  relations,  especially 


DISTINCTION    BETWEEN    UNITS   AND   QUANTITIES.    5 

in  substituting  one  kind  of  a  unit  for  another  in  a  formula,  which  will  be 
described  in  the  next  section. 

The  larger  the  unit  the  smaller  will  be  the  number  of  those  units  which  are 
contained  in  a  given  quantity;  that  is,  the  smaller  will  be  the  number  ex- 
pressing the  size  of  that  quantity  in  terms  of  those  units.  For  instance,  a 
meter  is  greater  than  a  yard,  hence  the  number  of  meters  in  a  given  distance 
is  less  than  the  number  of  yards.  This  self-evident  rule  is  often  of  value 
as  a  check  to  avoid  confusion  between  the  calculation  of  the  units  them- 
selves and  the  quantities  in  terms  of  those  units;  errors  of  this  kind  are 
especially  liable  to  occur  when  the  two  units  have  nearly  the  same  values. 

This  rule  should  not  be  confounded  with  another  quite  different  case  in 
which  the  values  of  two  different  units  are  given  in  terms  of  the  same  third 
unit;  for  instance,  1  kilogram -meter  =  2. 3  heat  units  and  1  foot-pound  =  0.32 
of  the  same  heat  units;  here  the  foot-pound  is  a  smaller  unit  than  the  kilo- 
gram-meter, hence  its  value  in  terms  of  a  third  unit  will  of  course  also  be 
smaller.  But  1  heat  unit  =  3.1  foot-pounds  and  1  heat  unit  =  0.43  kilogram- 
meter;  here  the  same  quantity,  1  heat  unit,  is  expressed  first  in  small  units, 
namely,  foot-pounds,  then  in  large  units,  kilogram-meters;  hence  3.1,  the 
number  of  the  smaller  units,  is  of  course  greater  than  0.43,  which  is  the 
number  of  the  larger  units. 

In  calculating  one  unit  from  another  unit,  great  care  must  be  taken  to 
avoid  multiplying  when  one  should  divide,  or  the  reverse.  Thus  if  the 
unit  1  kilogram-meter  equals  2.3  heat  units,  it  would  not  be  correct  to  say 
that  because  the  unit  1  kilogram  =  2. 2  Ibs  and  1  meter  =  3. 3  feet,  therefore 
the  unit  1  foot-pound  =  2. 3  X  (2. 2X3. 3)  =  16. 8  of  those  heat  units;  it  should 
be  1  foot-pound  =  2. 3 -4- (2.2X3. 3)  =  0.31  heat  units.  The  safest  way  to 
avoid  such  errors  is  to  remember  that  the  statement  1  kilogram-met er=  2.3 
heat  units  really  is  a  true  equation  between  units  and  means  1  kilogram  X  1 
meter  =  2. 3  heat  units;  one  should  then  substitute  for  1  kilogram  its  equal 
in  terms  of  the  other  unit,  namely  2.2X  1  pound,  and  for  1  meter  its  equal, 
namely  3.3X1  foot;  then  2.2X1  pound X 3.3 XI  foot  =  2. 3  heat  units,  which 
when  reduced  gives  the  unit  1  foot-pound  =  2. 3 -=-(2. 2X3. 3)  or  0.31  heat 
unit.  These  terms  here  always  refer  to  units  and  not  to  quantities  meas- 
ured in  terms  of  those  units,  which  latter  is  the  case  in  formulas. 

Expressing  this  relation  in  algebraical  terms,  it  means  A  X  B  =  2. 3C,  in 
which  A  is  one  kilogram  and  not  a  weight  denoted  in  kilograms,  B  is  one 
meter  and  not  a  distance  expressed  in  meters,  and  C  is  one  heat  univ ;  they 
should  here  be  considered  as  concrete  things  like  a  certain  piece  of  brass, 
a  certain  stick,  and  a  certain  lump  of  coal,  as  distinguished  from  abstract 
units  or  as  distinguished  from  a  mere  number  representing  a  measurement 
in  terms  of  those  units.  Now  if  in  the  same  way  a  is  one  pound  or  a  differ- 
ent piece  of  brass,  and  b  is  one  foot  or  a  different  stick,  then  as  1  kilogram  = 
2.2  pounds,  it  follows  that  ^4=2.2a  and  similarly  /?  =  3.36.  Substituting 
these  in  the  first  equation  gives  2.2aX3.3b  =  2.3C,  which  when  reduced 
gives  aX6  =  2.3-K2.2X3.3)C  =  0.31C. 

In  formulas,  however,  the  letters  do  not  stand  for  the  units  themselves, 
but  for  quantities  measured  in  terms  of  the  units,  and  hence  one  must  multi- 
ply instead  of  divide,  or  the  reverse,  PS  will  be  explained  in  the  next  section. 

A  similar  error  is  very  apt  to  arise  in  calculating  units  containing  the 
word  "per,"  which  word  signifies  a  division,  as  was  explained  above  under 
compound  units.  This  is  the  case,  for  instance,  in  finding  the  value  of  1 
kilogram  per  kilometer  from  the  value  of  1  pound  per  foot,  or  a  velocity 
of  1  mile  per  minute  from  that  of  1  foot  per  second.  It  must  be  remembered 
that  the  unit  following  the  word  "per"  is  a  divisor.  The  safest  way  always 
is,  as  above  described,  to  write  out  the  whole  expression  carefully,  then 
substitute  equivalent  values  and  reduce.  Thus:  1  mile  per  minute  =  26. 8 
meters  per  second,  may  be  written  1  mile  -*- 1  minute  =  26.8  meters  per  second ; 
then  to  find  the  value  of  1  foot  per  second  from  it,  substitute  5  280  feet  for 
1  mile,  and  60  seconds  for  1  minute,  thus  (5  280  X  1  foot) -J- (60  X  1  second)  = 
26.8  meters  per  second;  then  reduce  by  dividing  both  sides  by  5  280  and 
multiplying  both  by  60,  giving  1  foot-f-1  second,  or  1  foot  per  second  =  26.8 
-s-5  280X60  =  0.30  meter  per  second. 

When  the  unit  following  the  word  "per"  is  not  to  be  changed,  then  the 
division  just  referred  to  does  not  enter  into  the  calculation.  Thus  to 
reduce  some  value  of  1  foot  per  minute  to  that  of  1  mile  per  minute  or  to  that 
of  1  meter  per  minute,  is  merely  a  reduction  of  feet  to  miles,  or  to  meters. 


INTRODUCTION. 


REDUCING   FORMULAS   FROM  ONE  KIND   OF 
UNITS   TO  ANOTHER. 

It  often  occurs  in  practice  that  a  formula  is  given  for  one  set  of  units, 
such  as  meters,  kilograms,  seconds,  etc.,  and  it  is  desired  to  use  it  for  other 
units  such  as  .feet,  pounds,  minutes,  etc.  While  this  is  generally  a  simple 
calculation,  it  is  very  apt  to  be  made  incorrectly,  giving  an  entirely  wrong 
result.  The  error  arises  from  the  fact  that  one  is  apt  to  forget  the  distinc- 
tion between  a  unit  and  a  quantity  measured  in  terms  of  that  unit  (see  the 
preceding  section)  which  distinction  involves  the  difference  between  whether 
one  should  multiply  or  divide.  For  instance,  a  foot  is  larger  than  an  inch, 
but  the  number  of  feet  expressing  a  certain  distance  is  smaller  than  the  num- 
ber of  inches  expressing  that  same  distance. 

The  best  way  to  avoid  mistakes  is  to  remember  that  in  the  usual  formulas 
the  letters  represent  quantities  measured  in  terms  of  certain  units;  they  do  not 
represent  the  units  themselves.  To  change  a  formula  from  one  kind  of 
units  to  another,  it  is  therefore  necessary  to  replace  each  letter  of  the  original 
formula  by  another  letter  combined  with  such  a  reduction  factor  as  will 
reduce  the  measurement  made  in  terms  of  the  new  unit,  to  that  made  in 
terms  of  the  original  one.  Thus  if  a  formula  contains  the  letter  L  as  repre- 
senting length  expressed  in  meters,  and  it  is  desired  to  change  this  to  I 
expressed  in  feet,  then  substitute  for  L  the  equivalent  Z-j-3.28  or  ZX 0.305, 
because  any  length  I  measured  in  feet  when  divided  by  3.28  (or  multiplied 
by  0.305)  will  be  that  same  length  expiessed  in  meters;  and  as  the  original 
formula  is  correct  for  meters,  it  will  be  correct  to  substitute  IX 0.305  for  L 
because  the  number  thus  obtained  will  be  the  same  as  the  number  which 
expresses  L  in  meters.  Or,  if  the  original  formula  contains  the  letter  W, 
representing  weight  expressed  in  kilograms,  and  it  is  desired  to  change  it 
to  w  expressed  in  pounds,  then  substitute  for  W  the  equivalent  wX  0.454, 
because  any  weight  in  pounds  when  multiplied  by  0.454  will  give  the  same 
number  as  when  expressed  in  kilograms,  which  latter  number  is  what  the 
original  formula  called  for.  After  having  thus  replaced  each  letter  by  a 
new  one  and  a  constant,  all  these  constants  may  be  combined  into  one  if 
desired. 

It  must  not  be  forgotten  that  the  quantity  which  the  whole  formula  rep- 
resents, that  is,  the  quantity  which  is  to  be  calculated  by  means  of  the 
formula,  is  generally  also  in  terms  of  some  unit  (unless  it  is  a  mere  ratio, 
percentage,  or  number),  and  care  must  therefore  be  taken  to  also  substi- 
tute a  new  letter  and  reduction  factor  for  it,  if  it  is  desired  to  change  it  also. 
For  instance, if  a  formula  for  determining  the  metric  horse-power  from  data 
given  in  meters  and  kilograms,  is  to  be  changed  to  read  in  feet  and  pounds, 
as  just  described,  the  value  obtained  in  applying  the  new  formula  would 
still  be  in  metric  horse-power,  notwithstanding  that  the  formula  has  been 
reduced  to  feet  and  pounds;  if  the  result  is  to  be  in  English  horse-powers, 
then  the  letter  representing  the  metric  horse-power  must  likewise  be  changed 
to  a  new  one  with  its  reduction  factor,  just  as  was  clone  with  the  others. 
This  will  be  illustrated  below  by  an  example. 

The  safest  rule  of  thumb  for  the  changing  of  the  units  of  a  formula  is 
therefore  as  follows:  Substitute  for  each  letter  of  the  old  formula  a  new  letter 
multiplied  by  the  value  of  the  NEW  unit  in  terms  of  the  OLD  one,  as  given  in  the 
tables  in  this  book.  Then  the  new  letters  will  represent  quantities  meas- 
ured in  terms  of  the  new  units.  Thus  for  L  (meaning  meters)  substitute  I 
(meaning  feet)  X 0.305,  this  number  being  the  value  of  one  foot  (new  unit) 
in  terms  of  meters  (old  unit),  as  obtained  from  the  tables.  For  if  1  foot  = 
0.305  meters,  any  other  number  of  feet  (namely  Z)  must  be  multiplied  by 
0.305  to  reduce  it  to  the  number  of  meters  (L)  called  for  by  the  original 
formula;  or,  in  other  words,  L  and  IX 0.305  arc  equivalents,  and  either 
may  be  substituted  for  the  other. 

Or  put  into  algebraical  terms:  if  according  to  the  table  1  foot  =  0.305 
meter,  then  I  feet  are  equal  to  IX 0.305  meters;  or  a  length  represented  by  I 
in  feet  is  represented  by  IX 0.305  in  meters;  but  the  latter  quantity  is 
numerically  equal  to  I/,  hence  L  =  lX 0.305,  which  is  a  true  equation  in  which 
the  letters 'represent  the  same  length  measured  first  in  meters  and  then  in 
feet;  and  (IX 0.305)  may  therefore  be  substituted  for  L  in  any  formula. 


.     RATIOS. — PERCENTAGE.  7 

The  above  rules  are  illustrated  in  the  following  example:     Let 

in  which  P  represents  the  power  in  metric  horse-power  which  is  required 
to  operate  a  windlass  or  elevator  to  raise  W  metric  tons  L  meters  in  height 
in  T  seconds.  The  constant  16  includes  the  friction  loss,  and  the  reduction 
factors.  It  is  required  to  change  this  to  English  horse-powers  (p),  short 
tons  (w),  feet  (Z),  and  minutes  (/). 

From  the  tables,  one  English  horse-power  (the  new  unit)  is  equal  to  1.01 
metric  horse-powers  (the  old  unit),  hence  P  =  pXl.01.  One  short  ton  of 
2000  Ibs.  is  equal  to  0.907  metric  ton,  hence  JF  =  u?X0.907.  One  foot 
equals  0.305  meter,  hence  L  =  lX 0.305,  and  one  minute  equals  60  seconds, 
hence  T  =  <X60.  The  constant  16  being  in  terms  of  no  unit,  as  it  is  a  mere 
number  or  ratio,  remains  unchanged.  Substituting  all  these  in  the  old 
formula,  gives 

pxl.01  =  16<£2<-0-907x'x0-305 


which  whan  all  the  constants  are  combined,  gives  the  new  formula 
p  =  0.073-^, 

which  will  be  correct  for  quantities  expressed  in  terms  of  the  new  units. 

When  the  original  formula  is  to  be  applied  only  once  for  one  set  of  values 
in  other  units,  it  may  sometimes  be  simpler  to  reduce  the  original  data  in 
short  tons,  feet,  and  minutes  to  metric  tons,  meters,  and  seconds  by  means 
of  the  tables;  then  substitute  them  in  the  original  formula  and  calculate 
the  result,  which  will  be  in  metric  horse-power;  then  reduce  this  result  back 
to  English  horse-power  by  means  of  the  tables. 

RATIOS. 

A  ratio  means  the  relation  of  one  quantity  to  another,  that  is,  one  quan- 
tity divided  by  the  other.  It  is  therefore  always  a  mere  number  and  is 
independent  of  any  units,  except  that  the  two  quantities  concerned  must 
always  be  in  terms  of  the  same  units.  When  tho  statement  ;s  made  that  a 
certain  number  is  the  ratio  of  one  quantity  to  another,  it  invariably  means 
that  the  first  was  divided  by  the  second,  and  never  the  reverse.  A  con- 
venient rule  of  thumb  for  calculating  the  ratio  is  to  divide  by  the  one  that  is 
preceded  by  the  word  "to." 

Unless  otherwise  stated,  a  ratio  is  the  figure  thus  obtained  by  division 
and  when  it  is  reduced  to  a. single  number  (as  distinguished  from  a  vulgar 
fraction)  it  means  so-and-so  much  per  unity.  Thus  a  ratio  may  be  ex- 
pressed as  3  to  4  (or  %}  for  instance,  or  as  0.75;  in  the  latter  case  it  means 
0.75  per  unity.  Ratios  are,  however,  more  frequently  stated  as  percent- 
ages, that  is,  as  ratios  per  100,  in  which  case  the  above  defined  value  must 
be  multiplied  by  100.  Thus  the  ratio  of  3  to  4  is  then  expressed  as  75%, 
as  described  more  fully  below. 

PERCENTAGE. 

Percentage  values  represented  by  the  sign  %  and  meaning  per  hundred, 
are  often  used  as  a  convenient  way  of  representing  a  ratio,  fraction,  or  rela- 
tion, so  as  to  avoid  the  use  of  fractions,  at  least  in  most  cases.  Thus  a 
recovery  of  3  horse-power  out  of  4  may  be  expressed  as  the  fraction  %  or 
0.75,  or  it  may  more  conveniently  be  expressed  as  75%,  which  avoids  frac- 
tions and  reduces  the  relation  to  one  which  is  based  on  100.  The  objection 
to  it  is  that  such  values  of  necessity  introduce  a  multiplication  or  division 
by  100,  which  sometimes  gives  rise  to  confusion.  To  have  adopted  the 
system  of  stating  such  values  per  unity  instead  of  per  hundred,  would  have 
avoided  this  confusion,  but  the  values  would  then  in  most  cases  be  frac- 
tions, for  which  reason  the  percentage  values  are  usually  preferred. 

Percentage  values,  like  other  ratios,  are  mere  numbers  and  are  not  in 
terms  of  any  units ;  they  are  moreover  the  same  for  all  units  provided  only 
that  the  two  quantities  from  which  the  percentage  value  is  calculated  are 
always  in  terms  of  the  same  units. 


8  INTRODUCTION. 


While  percentages  occurring  in  practice  are  generally  between  1  and  100, 
yet  they  may  of  course  be  less  than  1  or  greater  than  100.  When  less  than 
1  care  should  be  taken  to  write  and  read  them  properly;  thus  .25%,  for 


percent."      Six  percen 

6%,  or  0.06,  but  never  0.06%. 

When  a  given  percentage  value  is  to  be  applied,  it  is  almost  always  the 
case  that  some  quantity  measured  in  units  is  to  be  multiplied  by  it.  In 
that  case  it  becomes  very  important  to  divide  either  the  percentage  value 
or  the  product  by  100,  in  order  to  get  the  actual  value  in  those  units.  Thus 
25%  of  8  is  0.25X8  =  2  or  (25X8) -^100  =  2.  Frequent  and  serious  errors 
arise  by  neglecting  this,  and  in  carelessly  retaining  the  percent  sign  (%) 
after  dividing  by  100,  by  writing  for  instance  0.06%  for  6%. 

When  the  percentage  value  is  to  be  found  from  two  given  quantities, 
errors  are  frequently  made  by  dividing  by  the  wrong  quantity.  Percentage 
va'ues  are  usually  expressed  by  saying  that  one  quantity  is  a  certain  per- 
cent of  another,  and  this  means  that  the  first  must  be  divided  by  the  second, 
and  then  multiplied  by  100.  A  good  rule  of  thumb  is  to  divide  by  the  one 
preceded  by  the  word  "of,"  and  then  multiply  by  100.  Thus,  2  is  what  per- 
cent of  8;  here  2  is  to  be  divided  by  8  and  multiplied  by  100,  giving  25%. 

Care  should  also  be  taken  to  reduce  the  original  figures  to  their  proper 
form  before  dividing  them  to  find  the  percentage.  Thus  to  find  what  per- 
cent is  6  less  than  8,  reduce  it  to  the  form  "what  percent  is  2  (that  is,  8 
minus  6)  of  8";  then  divide  2  by  8  and  multiply  by  100,  giving  25%.  Or, 
what  percent  is  8  greater  than  6;  divide  the  difference  2  by  6  and  multiply 
by  100,  giving  33^%. 

The  sign  °/oo  is  sometimes  used,  more  particularly  abroad ,  in  stating  grades. 
This  differs  from  percent  only  in  that  it  means  per  1  000  instead  of  per  100. 

CONDENSED  NUMBERS,     The  Use  of  1<K 

For  convenience,  brevity,  and  clearness  the  decimal  point  in  very  large 
or  very  small  numbers  is  often  moved  to  the  next  to  the  last  left-hand  digit, 
and  the  amount  by  which  it  has  thereby  been  lowered  or  raised  is  written 
separately  in  the  form  of  10  to  the  required  power.  Thus  the  large  num- 
ber 1230000  is  often  written  1.23  X  106  (that  is,  1. 23  X  1  000  000);  or 
0.00000123  is  written  1.23X10-6  (that  is,  1.23X0.000001).  A  positive 
exponent  of  10  indicates  that  the  real  number  is  larger  than  the  one  given, 
and  a  negative  exponent,  that  it  is  smaller.  The  exponent,  whether  posi- 
tive or  negative,  always  indicates  the  number  of  places  that  the  decimal 
point  has  been  moved.  When  the  exponent  is  negative  it  also  means  that 
in  the-original  number  there  is  one  less  zero  between  the  point  and  the  first 
digit  than  the  exponent  indicates;  thus  lfh~6  means  that  there  are  5  zeros 
between  the  point  and  the  first  figure;  this,  however,  applies  only  when 
the  decimal  point  in  the  condensed  number  is  between  the  first  and  second 
left-hand  digits,  as  usual;  sometimes,  though  not  according  to  good  prac- 
tice, 1. 23X10-°  is  written  12.3  X10~7,  in  which  case  this  rule  of  thumb 
does  not  apply. 

To  multiply  or  divide  such  condensed  numbers,  perform  that  operation 
with  the  numerical  part,  and  then  merely  add  (algebraically)  the  exponents 
of  the  10 's  in  the  case  of  a  multiplication  or  subtract  them  (algebraically) 
in  the  case  of  a  division.  Thus  (3.4  X  10ft)X  (5.6  X  104)  =  19.04  X  1010  or 
better,  1 .904X10".  Or  (3.4  X  10~6)X  (5.6  X  10~4)  =  19.04  X  10-'°,  or  better 
1.904X10-9.  Again,  (1.2X  106)X (3.4X  10~4)  =  4.08X  102;  or  (4.08X  102)-*- 
(3.4X10-4)  =  1.2X106,  as  -4  subtracted  from  +2  gives  +6 

To  square,  cube,  etc.,  such  a  condensed  number,  perform  that  operation 
with  the  numerical  part  and  multiply  the  exponent  of  10  by  2,  3,  etc., 
respectively.  For  extracting  the  square  or  cube  root,  perform  that  opera- 
tion with  the  numerical  part  and  divide  the  exponent  of  10  by  2,  3,  etc., 
respectively;  if  this  does  not  divide  evenly,  then  first  change  the  exponent 
so  that  it  will,  by  moving  the  decimal  point  in  the  numerical  part ;  thus,  to 
extract  the  cube  root  of  6.4 XlO7,  write  it  64.X106,  then  the  cube  root  is 
4.  X  102. 

To  add  or  subtract  such  condensed  numbers  they  must  first  be  reduced 
to  the  same  power  of  10;  then  add  or  subtract  the  numerical  parts  and 
affix  10  to  the  same  power.  Thus  (1.23X  104)  +  (4.56X  103)  =  (12.3X103)  + 


PREFIXES — APPROXIMATE    NUMBERS.  9 

(4.56  X103)  =  16.86  X103,  or  better,  1.686  X10*.  The  same  rule  applies 
when  all  the  exponents  are  negative,  or  when  some  are  positive  and  some 
negative,  although  in  the  latter  case  it  will  generally  be  simpler  to  reduce 
all  to  the  real  numbers  by  eliminating  the  factor  10. 

10°  is  neVer  used  in  such  numbers,  but  it  sometimes  results  from  calcula- 
tions.    It  is  equal  to  1,  and  is  therefore  always  omitted. 

TABLE  OF  CONDENSED    NUMBERS. 
Let  1.23  represent  any  number,  of  any  number  of  places  of  figures;  then: 


1.23  X106  .- 

1  230  000. 

1.23  X105    = 

123  000. 

1.23X104    = 

12  300. 

1.23X103    - 

1  230. 

1.23X102    = 

123. 

1.23  X101    = 

12.3 

1.23X10°    = 

1.23 

1.23X10-!  = 

0.123 

1.23X10-2  = 

0.012  3 

1.23X10-3  = 

0.001  23 

1.23X10-4  = 

0.000  123 

1.23X10-5  = 

0.000  012  3 

1.23X10~6  = 

0.000  001  23 

PREFIXES  used  in  the  METRIC  SYSTEM. 


Micro- 

0.000  001 

or 

10-6 

Milli- 

0.001 

or 

lO-3 

Centi- 

0.01 

or 

io-2 

Deci- 

0.1 

or 

10-1 

Deca-  or  Deka- 

10. 

or 

101 

Hecto-  or  Hekto- 

100. 

or 

IO2 

Kilo- 

1000. 

or 

IO3 

Myria- 

10  000. 

or 

10* 

Mega- 

1  000  000. 

or 

10« 

Thus  one  millimeter  equals  0.001  meter,  or  one  kilometer  equals  1  000 
meters. 

ACCURACY  OP  APPROXIMATE  OR  ABBREVIATED 
NUMBERS. 

When  a  value  is  known  to  be  correct  only  to  a  limited  number  of  places 
of  figures,  or  when  the  figures  are  abbreviated  to  a  few  places,  the  possible 
error  in  such  approximate  figures,  or  in  other  words  their  accuracy,  varies 
not  only  with  the  number  of  places  of  figures,  but  also  with  the  value  of  the 
left-hand  digit,  as  will  be  explained  below.  It  is  here  assumed  that  the 
last  right-hand  figure  has  always  been  increased  by  one,  when  the  next 
figure  would  be  5  or  over;  thus  anything  between  12.85  and  12.89  (includ- 
ing the  former)  would  be  abbreviated  to  12.9,  while  anything  between 
12.80  and  12.85  (but  not  including  the  latter)  would  be  abbreviated  to  12.8. 

The  possible  error  is  less  the  greater  the  number  of  places  of  figures.  For 
two  places  of  figures  (from  10  to  99)  the  greatest  error  (namely  ±  5  in  the  3d 
place)  is  from  5%  for  the  smaller  numbers  to  J^%  for  the  larger.  For  three 
places  of  figures  (100  to  999)  the  greatest  errors  are  *4%  to  ^0%-  For 
four  places  of  figures,  the  greatest  errors  are  ^0%  to  H>oo%-  For  five 
places  H>oo%  to  J4ooo%t  etc. 

The  possible  error  is  also  less  as  the  left-hand  digit  is  greater.  Thus  the 
abbreviated  number  10.1  may  mean  anything  from  10.05  to  nearly  10.15, 
and  the  greatest  error  may  therefore  be  ±5  in  the  next  (2d)  place  of  deci- 
mals, or  numerically  this  would  be  5  times  the  left-hand  digit  1.  But  in 
the  abbreviated  number  99.9,  which  may  mean  anything  from  99.85  to 
nearly  99.95,  the  greatest  error,  namely  ±5,  is  only  about  5/io  times  the 
left-hand  digit  9;  that  is,  in  percentage  only  about  Vio  as  great  an  error  as 
it  was  for  10.1.  It  follows  therefore  that  values  beginning  (at  the  left  hand) 
with  the  low  digits  1,  2,  3,  etc.,  should  be  stated  to  one  place  more  than 
those  beginning  with  the  high  digits  9,  8,  7,  etc.,  if  the  accuracies  of  the 


10  INTRODUCTION. 

abbreviated  values  are  to  be  more  nearly  the  same  for  all.  This  can  be 
made  use  of  to  advantage  in  tables  of  such  approximate  or  abbreviated 
values.  If,  for  instance,  such  a  table  is  intended  to  be  limited  to  three 
places  of  figures,  the  greatest  error  will  be  0.^%  for  the  1  w  numbers  (100) 
and  0.05%  for  the  high  ones  (999);  but  by  giving  f ;  ur  places  of  figures  for 
all  numbers  between  100  and  499,  and  three  places  for  those  from  500  to 
999,  making  an  increase  of  only  1  figure  for  every  6,  or  about  16%,  the 
greatest  error  will  be  reduced  fivefold,  namely  to  0.1%.  The  table  will  then 
take  a  mean  position  (in  accuracy)  between  a  throe-pk.ce  and  a  four-pi:  ,ce 
table;  the  variations  of  the  greatest  errors  in  diiierent  parts  of  the  table 
will,  however,  be  the  same  in  all  taree. 

The  location  of  the  .ecimal  point  Ices  not,  of  course,  affect  the  percentage 
accuracy;  thus  101.,  when  it  represents  an  approximate  or  abbreviated 
value,  is  just  as  accurately  stated  as  0.000  1:1  is. 

A  zero  (0)  at  tae  right-hand  end  of  a  whole  number  may  mean  either  a 
definite  known  quantity  like  a  digit  or  ar.  indefinite  unknown  one,  merely 
filling  a  vacant  pb.ce  of  figures;  there  is  unfortunately  no  way  of  distin- 
guishing between  them.  Thus  in  the  value  5280.  feet  to  the  mile,  the  zero 
might  either  mean  that  the  last  place  of  figures  is  exactly  correct  (which  is 
the  case  in  this  particular  value)  or  that  only  three  places  are  correct  and 
the  fourth  unknown.  In  a  decimal  fraction,  however,  a  zero  in  the  last 
place  is  always,  or  should  always  be,  understood  to  be  a  definite  known 
quantity;  thus  0.150  is  understood  to  be  known  or  correct  to  three  places, 
while  0.15  is  known  or  correct  to  only  two  places. 

The  following  table  gives  the  errors  in  abbreviated  numbers  in  a  more 
concise  form: 

Abbreviated  numbers.       Greatest  error.  Mean  probable  error. 

10  5.%  2.5% 

50  1.%  0.5% 

100  0.5%  0.25% 

500  0.1%  0.05% 

1  000  0.05%  25  parts  in  100  000 

5  000  0.01%  5  parts  in  100  000 

10  000  5  parts  in  100  000  25  parts  in  1  000  000 

50  000  1  part  in  100  000  5  parts  in  1  000  000 

100  000  5  parts  in  1  000  000  25  parts  in  10  000  000 

500  000  1  part  in  1  000  000  5  parts  in  10  000  000 

It  will  thus  be  seen  that  by  using  only  three  places  of  figures  to  represent 
any  data  or  results  (provided  the  last  right-hand  figure  has  been  raised  by 
unity  when  the  next  is  5  or  over)  the  greatest  possible  error  is  only  half  a 
percent,  and  the  probable  mean  error  is  about  Vio  to  %<>%•  Three  places 
of  figures  therefore  suffice  for  most  engineering  data. 

In  the  more  usual  simple  calculations  with  three-place  values,  the  mean 
probable  error  in  the  result  is  in  general  no  greater  than  this,  but  the  greatest 
possible  error  becomes  larger  and  may  affect  the  third  place  by  one  unit. 
Successive  multiplications  like  cubing  increase  the  error.  Hence  if  the 
result  is  to  be  quite  correct  to  three  places,  then  four  places  must  be  used 
to  start  with. 

ACCURACY  OF  LOGARITHMS. 

In  general,  the  number  of  (decimal)  places  in  the  logarithms  themselves 
should  be  one  greater  than  the  number  of  places  of  figures,  in  order  to  be 
as  accurate  (in  percent)  as  the  arithmetical  calculations  would  be.  The 
errors  are  therefore  easily  obtained  from  the  above  table  for  numbers.  Four- 
place  tables  are  therefore  sufficiently  accurate  for  three-place  figures. 


ABSOLUTE    SYSTEM. — C.  G.  S.  SYSTEM.  11 


ABSOLUTE  SYSTEM  of  UNITS.    The  O.  G.  S.  SYSTEM. 

Among  the  physical  quantities  there  are  some  that  are  fixed,  definite, 
independent  of  each  other,  and  invariable  all  over  the  universe,  while  others 
are  variable,  indefinite,  dependent,  arbitrary,  or  involve  empirical  con- 
stants. A  length,  for  instance,  belongs  to  the  former  class  and  is  invariable 
throughout  the  universe,  while  a  weight  considered  as  a  force,  is  different 
on  different  parts  of  the  earth  or  on  different  planets.  The  former  are 
for  the  sake  of  a  distinction,  called  absolute  quantities. 

Most  physical  quantities  are  dependent  on  others  or  are  derived  from 
others;  for  instance,  a  surface  is  the  product  of  two  lengths,  and  a  velocity 
is  a  length  passed  through  in  a  certain  time.  But  there  are  a  few  that  are 
independent  of  any  others  and  may,  therefore,  be  considered  to  be  funda- 
mental; a  length,  for  instance,  cannot  be  derived  from  anything  else.  By 
selecting  three  of  the  latter  from  among  the  absolute  quantities  most  and 
perhaps  all  of  the  other  physical  quantities  may  be  derived  from  them,  or 
defined  in  terms  of  them,  thus  making  a  single  uniform  system  in  which 
all  quantities  can  be  expressed  in  terms  of  some  function  or  combination 
of  these  three  fundamental  quantities.  When  a  definite  amount  of  each 
of  these  fundamental  quantities  is  taken  as  a  unit,  such  a  system  is  called 
an  absolute  system  of  units. 

Various  systems  of  this  kind  have  been  devised  differing  in  the  three 
quantities  which  have  been  selected  as  the  fundamental  ones.  The  most 
important  one  and  the  only  one  which  is  in  general  use,  is  based  on  the 
three  quantities,  length,  mass*  and  time,  usually  represented  by  the  letters 
L,  M ,  and  T  respectively,  or  I,  m,  and  t,  and  when  the  unit  amounts  are 
taken  as  the  centimeter  the  gram  (mass),  and  the  second  (mean  solar), 
the  system  is  called  the  centimeter-gram-second  system,  usually  denoted 
as  the  C.  G.  S.  system. 

In  this  system  the  units  of  each  of  the  derived  quantities  are  the  amounts 
which  correspond  to  the  unit  amounts  of  those  of  the  fundamental  quanti- 
ties which  are  involved.  Thus  the  unit  of  velocity  in  this  system  is  one 
centimeter  per  second;  the  unit  of  force,  called  a  dyne,  is  that  whi-;h  act- 
ing on  a  mass  of  one  gram  at  rest  produces  in  one  second  a  velocity  of  one 
centimeter  per  second;  the  unit  of  energy,  called  an  erg,  is  one  dyne  of 
force  acting  through  one  centimeter;  the  unit  of  electricity  or  magnetic 
pole  is  that  which  attracts  another  equal  amount  at  one  centimeter  dis- 
tance, with  a  force  of  one  dyne;  etc.,  etc. 

Sometimes  the  units  of  the  derived  quantities  depend  upon  how  they  are 
defined,  in  which  case  there  may  be  several  different  units.  The  most 
important  case  of  this  kind  is  that  of  the  electrical  units;  in  the  one  system, 
called  the  electrostatic  system,  a  whole  set  of  units  is  based  on  the  definition 
that  a  unit  of  electricity  is  that  which  attracts  an  equal  amount  one  centi- 
meter distant,  with  a  force  of  one  dyne,  while  in  the  other,  called  the  electro- 
magnetic system,  the  unit  current  is  defined  as  that  which,  flowing  through 
an  arc  of  one  centimeter,  curved  to  one  centimeter  radius,  generates  a  unit 
magnetic  pole  at  the  center;  this  definition  thus  connects  the  unit  of  elec- 
tricity with  the  unit  of  magnetism ;  on  it  the  electrical  units  in  common  use 
are  based.  The  relation  between  the  two  units  of  each  electric  quantity 
thus  defined  is  found  to  be  the  velocity  of  light  in  the  C.  G.  S.  system, 
or  some  power  of  it. 


*  Mass  represents  amounts  of  matter  and  must  be  clearly  distinguished 
from  weight.  A  given  mass  of  iron,  for  instance,  is  the  same  whether  it 
be  solid,  liquid,  volatilized,  oxidized,  dissolved,  etc.,  and  is  the  same  all 
over  the  universe,  while  its  weight  is  enormously  greater  on  the  sun  than 
-on  the  earth.  Two  given  masses  always  have  the  same  attraction  of  gravi- 
tation to  each  other  at  the  same  distance. 


12  INTRODUCTION. 

Sometimes  the  units  thus  defined  are  inconveniently  large  or  small  for 
practical  purposes,  and  therefore  certain  practical  units  have  been  chosen 
which  are  made  some  multiple  of  10  times  as  small  or  as  large  as  the  C.  G.  S. 
unit.  Thus  the  ampere  is  1/10  and  the  volt  is  one  hundred  million  times 
the  C.  G.  S.  unit  (electromagnetic)  of  current  arid  electromotive  force 
respectively.  The  definitions  of  all  these  units  and  the  relations  between 
them  are  given  in  their  respective  places  in  the  table  of  conversion  factors. 

While  most  physical  quantities  can  thus  be  deduced  from,  or  denned  in 
terms  of,  the  centimeter,  the  gram,  and  the  second,  yet  it  has  not  yet  been 
possible  to  do  this  with  all.  Among  the  important  exceptions  are  tem- 
perature, magnetic  permeability,  and  electric  inductive  capacity;  they 
should  therefore  be  added  to  the  fundamental  quantities  in  defining  or 
deducing  some  of  the  quantities.  For  most  purposes  the  two  latter  may 
be,  and  in  fact  are,  eliminated  from  consideration  by  making  them  equal 
to  the  numeral  1  or  unity  in  the  definitions.  They  are  therefore  called 
"suppressed"  fundamental  quantities. 

There  are  in  general  two  ways  of  establishing  units  of  various  quantities. 
One  way  is  to  define  an  absolute  unit  in  accordance  with  the  C.  G.  S.  system, 
and  then  establish  a  concrete  unit  in  terms  of  the  absolute  one,  which  will 
represent  the  latter  or  some  simple  decimal  multiple  of  it,  as  closely  as 
possible;  this  seems  to  be  the  more  rational  way,  as  it  maintains  the  whole 
system  uniform,  although  it  is  sometimes  difficult  to  establish  the  concrete 
unit  correctly;  this  is  the  method  adopted  in  establishing  the  electrical 
and  magnetic  units.  The  other  way  is  to  arbitrarily  establish  the  concrete 
unit  first,  by  selecting  some  convenient  standard,  and  then  determining 
experimentally  what  its  relation  is  to  the  natural,  absolute  unit;  this  is 
the  method  which  was  adopted  in  establishing  the  units  of  weight  (con- 
sidered as  a  force  and  not  as  a  mass),  heat,  temperature,  light,  etc.;  it  has 
the  disadvantage  that  the  relations  to  other  units  then  always  involve  an 
empirical  constant  which  is  necessarily  incommensurate,  like  the  accelera- 
tion of  gravity,  the  mechanical  equivalents  of  heat  and  of  light ;  but  it  has 
the  advantage  that  the  concrete  unit  is  established  definitely  at  the  start, 
and  is  not  subject  to  occasional  readjustment  like  the  concrete  electrical 
units. 


DIMENSIONAL  FORMULAS. 


When  any  physical  quantity  is  represented  algebraically  in  terms  of 
the  fundamental  quantities,  the  expression  is  called  its  dimensional  formula. 
Thus,  in  the  length,  mass,  and  time  or  L,  M,  and  T  system,  a  surface 
which  is  the  product  of  two  lengths  is  represented  by  L  X  L  or  L2,  a  volume 
by  L3;  these  are  the  dimensional  formulas  of  a  surface  and  a  volume,  and 
the  exponent  of  the  letter  is  called  its  dimension. 

It  frequently  happens  in  the  more  complex  formulas,  that  some  of  these 
letters  occur  as  divisors,  that  is,  as  the  denominator  of  a  fraction,  and  to 
avoid  stating  the  formula  in  terms  of  a  fraction,  the  exponents  are  made 
negative  in  such  cases.  Thus  a  velocity  which  is  length  divided  by  time, 
or  L/T,  is  generally  written  LT~ *;  an  acceleration  is  a  velocity  divided  by 
time  or  LT~l/T,  which  is  written  LT~ 2;  a  force  is  this  multiplied  by  mass 
or  L  M  T~2  and  energy  is  a  force  multiplied  by  a  length  or  L2  M  T~2,  etc. 
Fractional  exponents  denote  roots,  thus  La  means  the  square  root  of  L; 
or  L~~i  denotes  the  cube  of  the  square  root  of  L  in  the  denominator,  etc. 

In  some  cases  the  letters  cancel  each  other,  leaving  the  exponent  0;  the 
dimensional  formula  is  then  simply  1,  sometimes  called  a  number;  an  angle, 
for  instance,  is  defined  as  an  arc  divided  by  the  radius,  that  is  Z/-5-Z/  =  Z/°  =  1; 


DIMENSIONAL   FORMULAS.  13 

the  same  is  true  of  an  efficiency  which  is  energy  divided  by  energy,  or  power 
divided  by  power,  etc. 

When  intelligently  interpreted  and  applied,  such  dimensional  formulas 
are  often  very  useful.  They  frequently  give  an  idea  of  the  physical  nature 
of  a  quantity,  and  more  particularly  of  its  relation  to  other  quantities; 
they  sometimes  show  that  several  quantities  which  have  been  defined  in 
entirely  different  ways,  and  which  originated  differently,  are  really  the 
same;  they  sometimes  point  out  the  existence  of  some  law;  they  aid  one 
in  determining  what  the  rational  unit  is,  of  a  quantity  for  which  no  unit  has 
been  chosen;  they  are  useful  for  finding  whether  a  physical  formula  is 
correct  and  complete;  these  are  only  a  few  of  the  uses  of  such  formulas. 
The  quotient  of  two  such  formulas  often  shows  that  the  relation  between 
the  quantities  is  a  third  simple,  well-known  quantity.  Thus  the  relation 
between  energy  and  pressure  is  a  volume,  or  the  relations  between  the 
formulas  for  the  electrical  units  defined  electrostatically,  and  those  defined 
electromagnetically,  is  in  this  way  shown  to  be  the  velocity  of  light  or  its 
square,  which  is  one  of  the  foundations  of  Maxwell's  electromagnetic  theory 
of  light. 

It  must  not  be  forgotten  that  the  dimensional  formulas  usually  used  are 
based  on  the  fundamental  units,  length,  mass,  and  time,  and  that  they 
would  be  quite  different  if  other  fundamental  quantities  are  used ;  they  are 
therefore  only  relative  and  not  absolute,  and  their  chief  use  is  therefore 
to  show  the  relations  between  different  quantities  rather  than  the  physical 
nature  of  any  one  considered  by  itself. 

Although  dimensional  formulas  are  very  useful,  great  caution  should  be 
exercised  in  applying  them,  as  inconsistencies  and  absurdities  may  result 
if  they  are  treated  as  mere  algebraic  formulas.  Energy,  for  instance,  is 
force  X  length,  which  gives  the  formula  L2  M  T~2,  but  torque  is  also  force  X 
length,  and  therefore  has  the  same  formula,  although  it  is  physically  an 
entirely  different  quantity.  When  torque  acts  through  an  angle  it  becomes 
energy,  hence  torque  X  angle  =  energy,  but  as  the  dimensional  formula  of  an 
angle  is  1  the  formula  for  torque  is  not  changed  by  multiplying  it  by  an 
angle.  The  length  involved  in  torque  is  perpendicular  to  the  force,  while 
in  energy  it  is  in  the  same  direction,  therefore  the  two  lengths  have  a  differ- 
ent physical  meaning  in  the  two  cases,  as  they  are  at  right  angles  to  each 
other;  as  this  cannot  be  indicated  in  the  formula,  it  shows  that  the  system 
is  defective.  The  "angle"  in  such  a  case  has  been  termed  a  "suppressed 
quantity"  in  the  formula,  and  it  is  due  to  such  quantities  that  errors  may 
arise  in  using  dimensional  formulas.  Its  real  existence  in  the  formula  for 
torque  should  at  least  be  indicated  by  some  auxiliary  letter  like  a,  for  in- 
stance, calling  attention  to  its  suppression  in  the  rest  of  the  formula. 

It  seems  to  have  been  impossible  so  far  to  find  the  true  dimensional  formula 
of  temperature,  or  even  to  determine  whether  one  exists;  it  is  therefore 
safest  at  present  to  consider  it  an  auxiliary  fundamental  quantity,  usually 
represented  by  6,  and  add  it  to  the  formula.  Further  remarks  on  this  sub- 
ject will  be  found  in  a  footnote  under  the  thermal  units  in  the  table  of 
physical  quantities  given  below. 

In  the  formulas  for  the  electrical  and  magnetic  units  there  are  also  some 
"suppressed"  quantities  similar  in  some  respects  to  the  angle  above  men- 
tioned; like  in  the  case  of  temperature,  their  dimensional  formulas  are 
not  known,  they  are  generally  omitted  in  the  formulas  for  other  derived 
quantities,  but  without  them  these  formulas  are  not  complete  and  may 
mislead.  Ordinarily,  they  are  considered  to  be  unity  and  therefore  are 
algebraically  eliminated  from  the  formulas,  but  as  was  shown  above  con- 
cerning the  quantity  "angle,"  this  may  lead  to  misconceptions.  Until 
their  dimensional  formulas  are  known  it  is  recommended  to  consider  them 
as  auxiliary  fundamental  quantities  and  to  add  to  the  formulas  a  symbol 
which  represents  them,  in  order  to  call  attention  to  the  fact  that  they  really 
exist  but  are  suppressed  in  the  particular  system  used.*  In  the  dimensional 
formulas  given  below  in  the  table  of  physical  quantities,  they  have  been 


*  Prof.  Ruecker,  in  the  paper  referred  to  below,  says:  "I  think  the  sym- 
bols are  thus  made  to  express  the  limits  of  our  knowledge  and  ignorance 
on  the  subject  more  exactly  than  if  we  arbitrarily  assume  that  some  one  of 
*,he  quantities  is  an  abstract  number." 


14  INTRODUCTION. 

added  in  parentheses;   for  ordinary  purposes  they  may  be  considered  as 
being  unity.     These  quantities  are  the  electric  inductive  capacity  repre- 


formulas  should  be  unity;  a  definition,  for  instance,  might  be  based  on  a 
unit  distance,  but  it  would  be  quite  wrong  to  therefore  leave  out  of  the 
dimensional  formula  the  L  which  represents  it. 

Energy  being  always  the  same  quantity  in  all  systems,  never  has  any 
of  these  suppressed  factors  in  its  formula;  the  same  is  true  of  power. 

Although  the  dimensional  formulas  of  these  two  suppressed  quantities 
individually  are  unknown,  that  of  both  combined  is  known  in  the  follow- 
ing form: 


in  which  v  is  the  velocity  of  light  in  the  C.  G.  S.  system ;  hence  when  the 
formula  of  one  of  them  is  known,  that  of  the  other  can  be  determined. 
It  was  thought  by  Williams  (see  reference  below)  that  fi  may  be  a  density. 

The  dimensional  formulas  of  the  photometric  quantities  have,  it  seems, 
never  been  given.  Those  in  the  following  table  are  suggested  by  the  author. 

A  concise  discussion  of  the  deduction  of  the  dimensional  formulas  of 
many  of  the  usual  physical  quantities  will  be  found  in  the  Smithsonian 
Physical  Tables,  edited  by  Prof.  Thomas  Gray,  2d  edit.  p.  xv  to  4.  The 
"suppressed"  quantities  are  discussed  in  a  paper  by  Ruecker  in  the  Phil. 
Mag.,  Feb.  1889,  vol.  27,  p.  104,  supplemented  by  another  by  Williams, 
Phil.  Mag.,  1892, p.  234.  The  table  of  physical  quantities  given  below  con- 
tains all  the  quantities  whose  formulas  are  given  in  these  references  besides 
numerous  others. 

DECISIONS  OF  INTERNATIONAL  ELECTRICAL 

CONGRESSES 

Concerning   Electric,    Magnetic,   and  Photometric  Units  and 
Definitions. 

THE  following  is  a  brief  summary  of  these  decisions,  adoptions,  and 
recommendations.* 

The  official  congress  of  1881  in  Paris  adopted  the  fundamental  units 
centimeter,  gram  (mass),  and  second,  for  the  electric  measures;  the  ohm 
as  equal  to  109  C.  O.  S.  units  t;  the  volt  as  equal  to  10s  C.  G.  S.  units  f;  the 
ampere  as  the  current  produced  by  a  volt  through  an  ohm;  the  coulomb 
as  the  quantity  corresponding  to  an  ampere  for  one  second;  the  farad  as 
the  capacity  corresponding  to  a  charge  of  a  coulomb  by  a  volt.  It  in- 
structed an  international  commission  to  determine  what  the  length  of  a 

in  order  to  have  a  resistance  of  one  ohm  as  above  defined;  the  commission 
reported  in  1884  that  this  length  was  106  centimeters  and  called  this  ohm 
the  legal  ohm;  it  also  recommended  that  this  be  adopted  internationally; 
it  also  recommended  that  the  ampere  be  made  equal  to  10-1  C.  G.  S. 
(electromagnetic)  units,  and  that  the  volt  be  that  electromotive  force 
which  maintains  a  current  of  one  ampere  (presumably  as  just  defined) 
through  a  resistance  of  one  legal  ohm;  (this  volt  has  since  often  been  re- 
ferred to  as  the  legal  volt,  although  there  seems  to  be  no  legal  sanction 
for  this  name).  The  congress  of  1881  also  recommended  that  an  inter- 
national commission  be  appointed  to  define  a  standard  of  light;  this  com- 
mission reported  in  1884  that  the  unit  of  each  kind  of  simple  light  be 

*  For  a  more  detailed  summary  up  to  1900  see  Recapitulation  des  Deci- 
sions des  Conyres  Anterieurs,  by  Hospitalier,  in  the  report  entitled  "Con- 
gres  International  d'Electricite/'  1900,  pages  11-22;  for  the  adoptions  of 
the  congress  of  1900  see  pages  369  and  370  of  that  report. 

f  The  electromagnetic  system  was  unquestionably  meant,  and  is  under- 
stood to  be  meant  in  all  that  follows  here. 


DECISIONS    OF    CONGRESSES.  15 

the  quantity  of  light  of  the  same  kind  emitted  perpendicularly  from  a  square 
centimeter  of  surface  of  melted  platinum  at  the  temperature  of  its  solidi- 
fication; and  that  the  practical  unit  of  -white  light  be  the  total  light 
thus  emitted.  (The  name  violle  has  since  come  into  use  for  this  unit, 
although  apparently  without  official  adoption.) 

The  official  congress  of  1889  in  Paris  adopted  the  joule  as  equal  to  107 
C.  G.  S.  units  of  work  (ergs)  and  defined  it  also  as  the  energy  represented 
per  second  by  one  ampere  through  one  ohm;  (the  ohm  here  referred  to  is 
presumably  that  defined  in  terms  of  the  absolute  system,  and  not  the  "legal " 
ohm);  the  watt  as  equal  to  107  C.  G.  S.  units  of  power  (ergs  per  second), 
and  therefore  equal  to  a  joule  per  second;  the  kilowatt  as  the  industrial 
unit  of  power  in  place  of  the  horse-power;  the  bougie  dec! male  (decimal 
candle)  for  the  practical  unit  of  light  as  equal  to  the  twentieth  part  of  the 
absolute  standard  of  light  defined  by  the  commission  in  1884  (namely,  the 

Elatinum  standard  described  above  and  called  the  violle);  the  quadrant 
:>r  the  practical  unit  of  self-induction  (now  called  inductance)  as  equal 
to  109  centimeters;  the  period  of  an  alternating  current  was  defined  to 
be  the  duration  of  one  complete  oscillation,  and  the  frequency  the  num- 
ber of  periods  per  second;  the  mean  intensity  (of  a  current)  was  defined 
by  an  algebraic  expression  which  signifies  the  arithmetical  average  of  all 
the  instantaneous  values;  the  effective  intensity  (of  a  current)  was 
defined  to  be  the  square  root  of  the  mean  square  of  the  intensity  of  the 
current,  and  the  effective  electromotive  force,  the  square  root  of 
the  mean  square  of  the  electromotive  force ;  the  apparent  resistance  (now 
called  impedance)  was  defined  to  be  the  factor  by  which  the  effective  in- 
tensity of  the  current  must  be  multiplied  to  give  the  effective  electromo- 
tive force;  the  positive  plate  of  an  accumulator  was  defined  to  be  that 
which  is  the  positive  pole  during  discharge. 

The  unofficial  congress  of  1891  at  Frankfort  (Germany)  decided  *  that 
all  units  be  expressed  in  Roman  type,  all  physical  quantities  in  italics,  and 
all  physical  constants  and  angles  in  Greek  type;  also  that  the  quantities 
ampere,  coulomb,  farad,  joule,  ohm,  volt,  and  watt  be  expressed  by  their 
initial  letters,  A,  C,  F,  J,  O,  V,  and  W.  (These  decisions  were  not  con- 
firmed by  the  subsequent  official  congress,  and  have  not  come  into  general 
use.) 

The  official  congress  of  1893  in  Chicago  f  adopted  the  international 
ohm,  based  on  the  ohm  equal  to  109  C.  G.  S.  units  and  represented  by  the 
resistance  of  a  column  of  mercury  at  0°  C.,  106.3  centimeters  long,  weigh- 
ing 14.452  1  grams,  and  having  a  uniform  cross-section;  (this  cross-sec- 
tion, although  not  so  stated  officially,  is  practically  one  square  millimeter); 
the  international  ampere  as  equal  to  lO"1  C.  G.  S.  units,  represented 
for  practical  purposes  by  the  current  which  will  deposit  0.001  118  gram  of 
silver  per  second;  the  international  volt  as  that  electromotive  force 
which  will  maintain  one  international  ampere  through  one  international 
ohm,  represented  for  practical  purposes  by  1-4- 1.434  of  that  of  a  Clark  cell 
at  15°  C. ;  the  international  coulomb  as  the  quantity  corresponding  to 
one  international  ampere  in  one  second;  the  international  farad  as 
corresponding  to  a  charge  of  one  international  coulomb  by  one  interna- 
tional volt;  the  joule  as  equal  to  107  C.  G.  S.  units,  and  represented  in 
practice  by  the  energy  in  one  second  of  an  international  ampere  passing 
through  an  international  ohm ;  the  watt  as  equal  to  107  C.  G.  S.  units  and 
represented  in  practice  by  one  joule  per  second;  the  henry  as  "the  induc- 
tion in  a  circuit  when  the  electromotive  force  induced  in  this  circuit  is  one 
international  volt,  while  the  inducing  current  varies  at  the  rate  of  one 
ampere  per  second'-'  (presumably  meaning  the  international  ampere). 
These  eight  units,  substantially  as  defined  by  this  international  congress, 
were  made  legal  in  the  United  States  by  Act  of  Congress  in  1894.  For 
practical  magnetic  units,  the  C.  G.  S.  units  were  commended  by  the  Chicago 
congress,  but  no  names  were  given  them;  a  report  of  a  committee  on 
notation  and  nomenclature  was  received  and  ordered  printed  as  an  appen- 

*  See  Electrical  World,  vol.  18,  1891,  page  248. 

t  For  further  details  see  page  20  of  the  Proceedings  of  the  International 
Electrical  Congress  held  at  Chicago,  1893,  published  by  the  Amer.  Inst. 
Elect.  Engineers. 


INTRODUCTION. 


dix,  no  further  action  being  taken  on  it;  (see  the  table  of  physical  quanti- 
ties below,  in  which  that  report  is  included). 

The  unofficial  congress  of  1896  in  Geneva  adopted  the  bougie  decimale 


angle;  the  lux  as  the  unit  01  illumination  [E]  equal  to  one  lumen  per  square 
meter;  the  bougie  per  square  centimeter  as  the  unit  of  brightness  [e\\ 
the  lumen-liour  as  the  unit  of  quantity  of  light  [O]. 

The  official  congress  of  1900  in  Paris  adopted  the  name  gauss  for  the 
C.  G.  S.  unit  of  intensity  of  magnetic  field  or  flux  density,  and  the  name 
maxwell  for  the  C.  G.  S.  unit  of  magnetic  flux. 

For  the  values  of  these  official  units  in  terms  of  each  other  and  of  other 
units,  see  the  respective  tables  of  measures. 


PHYSICAL    QUANTITIES    AND    RELATIONS.          17 


TABLES  of  PHYSICAL  QUANTITIES  and  RELATIONS. 

The  following  table  gives  the  physical  quantities  and  relations  in  use, 
with  their  names,  symbols,  derivation,  dimensional  formulas  in  the  C.  G.  S. 
system,  and  the  units  whenever  such  units  have  been  generally  adopted. 
A  similar  though  much  smaller  table,  limited  chiefly  to  the  more  important 
electrical  and  magnetic  quantities,  was  recommended  by  the  Committee 
on  Notation  of  the  International  Electrical  Congress  of  1893  in  Chicago;* 
this  has  been  included  here  after  revision,  correction,  and  some  rearrange- 
ment. The  "suppressed"  factors  k  and  «  have  been  added,  which  neces- 
sitated some  changes  in  the  derivational  formulas  of  the  Congress  table, 
notably  that  of  magnetic  flux  which  now  becomes  BS  instead  of  HS.  The 
dimensional  formulas  then  become  consistent  throughout;  moreover,  they 
then  represent  the  conditions  of  practice  better,  as  iron  is  used  in  nearly 
all  forms  of  magnets. 

The  magnetic  quantities  based  on  the  electrostatic  system,  which  are 
riot  generally  given  in  text-books,  have  here  been  added  for  the  sake  of 
completeness. 

For  the  definitions  of  the  units  and  the  quantitative  relations  between 
them  see  the  respective  tables  of  measures. 


*  Reprinted  with  some  additions  in  the  "Electrical  World  and  Engineer," 
vol.  37,  January  5,  1901,  p.  501. 


18 


INTRODUCTION. 


PHYSICAL  QUANTITIES  AND  RELATIONS. 


Name. 

Sym- 
bol. 

Derivation. 

Dimensional 
Formula. 

'C.  G.  S.  Unit.* 

Fundamental. 

L,l 

L 
M 
T 

d 
k 
n 

L2 
£3 

number 
number 
L-i 
L-2 

L-3M 

number 
LMT-* 
LT-z 

L*M-iT~* 

LST~* 
MT-z 

centimeter 
gram  (mass) 
second  (mean 
solar) 

1  sq.  centimeter 
cubic   centimeter 
radian 

1  sq.  cm  at  1  cm 
radius 

gram  (mass)  per 
cb.  cm 

1 
dyne 
dyne  per  gram 

dyne  per  centi- 
meter 

Mass  .  . 

M 
T,t 

Time  

Auxiliary     funda- 
mental quantities. 
Temperature  
Electric  inductive 
capacity              * 

0 
k 
H 

S,s 
V 

a,/? 

Magnetic   permea- 
bility 

Geometric. 

Surface  

Volume.  .  . 

LL 

LLL 
arc 

Angle,  plane  
Angle,  solid  

radius 
spherical  area 

Curvature  or  Tor- 

[.... 

}•- 

d 
sp.    gr. 

("' 

(.:.. 

}::;• 

.  radius2 
angle 

Specific  curvature 
of  a  surface  

Mechanical. 

Weight:   see  Mass 
and  Force. 
Density.  .  . 

length 
solid  angle 

surface 

M 
V 

density 

Specific  gravity.  .  . 
Force. 

density 
Ma 
F 

M 

FL* 
M2 

FL* 
M 

F 
L 

Intensity  of  attrac- 
tion or  force  at  a 
point.  . 

Gravitation     con- 
stant   

Force  of  a  center 
of   attraction   or 
strength  of  a  cen- 
ter   

Surface  tension.  .  . 

*  For  the  names  and  abbreviations  of  the  numerous  practical  units  and 
for  their  values  in  the  C.  G.  S.  units,  see  the  various  tables  of  measures. 


PHYSICAL   QUANTITIES    AND    RELATIONS. 


19 


PHYSICAL    QUANTITIES  AND   RELATIONS  (Continued). 


Name. 

Sym- 
bol. 

Derivation. 

Dimensional 
Formula. 

C.  G.  S.  Unit. 

Pressure  or  inten- 
sity of  stress.  .  .  . 

P 

F 

S 

L-.MZ- 

barie* 

Modulus    of    elas- 
ticity 

IF 

L—  1MT~  2 

Resilience.  .  . 

LS 
W 

L~1MT~2 

Torque,    moment, 
couple 

V 
FLor       W 

L2MT~2  1 

angle 

yne  cen  ime  er 

Directive  force  (as 

\ 

torque 

in  suspensions).  . 

Fv 

angle 

L2MT~2 

Moment  of  inertia. 
Inertia  

K 

MX  radius2 

L2M 
LM 

gm  (mass)  cm  sq. 

L 

Velocity,  linear.  .  . 

V 

L 
T 

LT~< 

centimeter  per 
second;  kine 

Velocity,  angular 

0) 

angle 

T-i 

radian  per  second 

Acceleration,    lin- 
ear     .  . 

T 

V 

LT~2 

centimeter  per 

T 

sec.  per  sec. 

Acceleration,     an- 
gular. . 

<0 

T-2 

radian     per    sec. 

Momentum     or 
quantity  of  mo- 
tion. . 

[.... 

T 

mass  X  veloc- 
ity 

i 
LMT-i 

per  sec. 

Moment     of     mo- 
mentum or  angu- 
lar momentum  .  . 

\- 

momentum  X 
length 

»MT* 

Energy,  work.  .  .  . 
Vis-  viva.  .  . 

W 
W 

FL 

L2MT~2 
L2MT~2 

erg 
erg 

Impact.  . 

w 

L2MT~2 

erg 

Power  or  activity.. 

P 

W 
T 

erg  per  second 

Efficiency  

power 

number 

1% 

power 

*  It  equals  a  dyne  per  sq.  centimeter.     This  name  has  not  yet  been  defi- 
nitely adopted ;   some  use  it  for  the  pressure  of  one  atmosphere. 

t  More  correctly  this  should  also  contain  the  reciprocal  of  an  angle. 


20 


INTRODUCTION. 


PHYSICAL   QUANTITIES  AND    RELATIONS  (Continued) 


Name. 

Sym- 
bol. 

Derivation. 

Dimensional 
Formula. 

C.  G.  S,  and 
Practical  Units. 

Magnetic. 

(El'mag.  system.) 

Reluctance  ormag- 
netic  resistance.  . 

(R 

L_ 

Hr') 

oersted  * 

Reluctivity  ;     spe- 

cific reluctance. 

V 



number  X(f*~l) 

Permeance  

1 

J 

(R 

Magnetic  capacity 

C 

1 

(R 

UA 

Permeability       or 

specific       induc- 
tive capacity  .  .  . 

„ 

(B 

number  X(/*) 

Susceptibility.  .  .  . 

K 

3 

5C 

number  X(  At) 

Magnetomotive 
force     .  • 

ff 

4nNn 

L^M^T~l(fi~^) 

gilbert  T 

Magnetic      poten- 

W 

tial  

L^M^T~~l(fj~^) 

m 

Magnetizing  force. 

3C 

~L~I 

L~^M^T~l(n~^) 

gilbert  1  per  cen- 
timeter;  gauss 

Field  intensity.  .  . 

JC 

F 

m 

L~~b  M  %  T~l  (  fT~%  ) 

gauss 

Flux  density  

(B 

~~S 

L—  $M$T—l(ffc) 

gauss 

Magnetic   indue- 

tion.  . 

(B 

L—%M?T-l([jp) 

gauss 

Intensity  of  mag- 

arc 

netization  

3 

L~^M^T~1(fi^f) 

~V 

Flux  or  magnetic 
lines  of  force  .... 

0 

&s 

L$M^T~l(t&) 

mixwell 

Strength  of  pole  or 

) 

quantity  of  mag- 

\m 

\/L2Ffji 

L*M%T~l([jfc) 

netism  

) 

Magnetic  moment. 

9TC 

ml 

WMt^-i(^t) 

Magnetic  energy... 

W 

03 

Lwr-2 

erg 

Magnetic  power.  . 

P 

Jg| 

Lwr-a 

erg  per  second 

*  Provisionally  adopted 
t  N=:  number  of  turns. 
I"  Provisionally  adopted 
unit  is  the  ampere-turn. 


by  the  Amer.  Inst.  of  Electrical  Engineers. 

t  L  =  length  of  coil.  §  n  =  frequency, 

by  the  Amer.  Inst.  Elec.  Engineers.     The  usual 
1  gilbert  =  0.795  8  ampere-turn. 


PHYSICAL    QUANTITIES  AND    RELATIONS. 


21 


PHYSICAL   QUANTITIES  AND   RELATIONS    (Continued). 


Name. 

Sym- 
bol. 

Derivation. 

Dimensional 
Formula. 

Ratio  of 
Electro- 
static to 
Elec'mag- 
n'tic  Units* 

Magnetic. 

(Electrostatic  system.) 

Reluctance  or  magnetic 

(R 

& 

V2 

Reluctivity;  specific  re- 
luctance. .  .  . 

1 

V2 

Permeance. 

A* 

1 

L~lT2(k~l) 

v~2 

Magnetic  capacity  

Permeability  or  specific 
inductive  capacity.  .  .  . 

Susceptibility.  ....'.... 

€ 

K 

(R 
1 
(R 
ffl 

3C 
3 

v-2 

Magnetomotive  force.  .  . 
Magnetic  potential.  .  . 

3C 
$(${ 

L^M^T~2(k^) 
L%M%T~2(kb) 

V 

Magnetizing  force.  .  . 

5C 

l$M%T~2(k^) 

Field  intensity.  .  . 

3C 

L 
F 

Flux  density. 

m 
(P 

L-%M%(k-^) 

Magnetic  induction  

Intensity  of  magnetiza- 
tion 

3 

S 
~~S 

L—*M%(k~^) 

,-. 

Flux  (or  magnetic  lines 
of  force)  . 

f 

V 
ET 

/*M>«-*> 

Strength  of  pole  or  quan- 

W 

T  ^  Jl/T^f  1*  —  ^t\ 

Magnetic  moment  
Magnetic  energy.  . 

W 

magn.poten. 
ml 

L2MT  —  2 

1 

Magnetic  power.  ... 

p 

W 

L2MT  —  ^ 

I 

T 

*  vis  the  velocity  of  light  in  the  C.  G.  S.  system,  namely  3X1010  centi- 


22 


INTRODUCTION. 


PHYSICAL  QUANTITIES    AND   RELATIONS    (Continued). 


Name. 

Sym- 
bol. 

Derivation. 

Dimensional 
Formula. 

Practical  Units.* 

Electric. 

(  El'  mag.  system.) 

p 

Resistance.  .  . 

R.  rt 

LT~*(  «) 

ohm 

Resistivity  or  spe- 
cific resistance.    . 

P 

RS 
L 

L2T~l(n) 

ohm-centimeter 

X,  x  f 

I 

hm 

Magnetic       react- 

2nnC 

ance  

X 

ohm 

Capacity       react- 
ance or  condens- 
ance   .  . 

I 

Lr-'(,> 

ohm 

Impedance        . 

Z,  g\ 

V^T^2 

L 

ohm 

Conductance  

G,g1[ 

1          r 

L"~177(/<~1) 

mho 

Conductivity  or 
specific  conduct- 
ance  .  .        ... 

\> 

1 

P 

*+«$ 

mho  per  centi- 
meter 

Admittance   .  . 

Y  y\ 

1        1 
—  or  — 

L-iT(  -i) 

mho 

Z       z 

Susceptance.  .... 

B.6t 

X/^? 

L-iT(^-0 

mho 

Electromotive 

force  

E,  e 

2 

L%M^T~2(  1$  ) 

volt 

T 

Potential 

W 

z,itf*r-*(A£*) 

volt 

Difference    of   po- 

Q 

tential            .... 

U,u 

RI 

L^M^T~2(fi^) 

volt 

Electromotive 

E 

force  at  a  point  . 

~L 

L^M^T~2(n^) 

volt 

Intensity  of  elec- 

F 

, 

tric  field  

L%M%T-2(fj?} 

~Q 

ET 

Vector  potential  . 

~J7 

L^M^T~l(fj^} 

Current  

M 

E 
R 

L^M^T~l(fj~^) 

ampere 

Current  density 

I 

L~i  M^  T~l  (  u~^  ) 

ampere  per  sq.  cm 

S 

Quantity   of   elec- 
tricity; charge. 
Surface  density  or 

Q,Q 

IT 

L^M^djT  ^) 

coulomb  ;  am- 
pere-hour 

electric  displace- 

f 

_*_ 

L-$Mb(n~b') 

coulomb   per   sq. 

ment 

S 

cm 

*  The  C.  G.  S.  units  have  no  names.     For  the  relations  between  the  values 
of  these  units  and  the  C.  G.  S.  units,  see  the  various  tables  of  measures. 

*  Vector  quantities  when  used  should  be  denoted  by  capital  italics. 


PHYSICAL   QUANTITIES   AND    RELATIONS.         23 


PHYSICAL   QUANTITIES  AND   RELATIONS   (Continued*,. 


Name. 

Sym- 
bol. 

Derivation. 

Dimensional 
Formula. 

Practical  Units. 

Capacity  

C,  c 

Q 

L~lT2(/j~l) 

micro-farad 

Electric  inductive 
capacity,  or  Di- 
electric constant, 
or  Specific  induc- 
tive capacity.  .  .  . 
Inductance  or  co- 
efficient  of    self- 
induction  or  elec- 
tro-kinetic   iner- 
tia   

|»| 

j          I 

MM 

E 
C 
L 

c.^<L 

N3>* 

I 

number 

henry 

Mutual  inductance 

L(n) 

henry 

Inductance  factor  : 
see  below. 
Tim*1  constant 

L 

T 

second  ;    henry 

Period 

J_ 

T 

per  ohm 
second 

Frequency  

n 

n 

periods  per  sec. 

Angular  velocity.. 
Electro-kinetic 
momentum  
Thermoelectric 
height  or  specific 
heat  of  electricity 

Coefficient  of  Pel- 
tier effect  

Ditto 

\   ' 
|   .... 

litn, 

/Xinductance 
E 

heat 
IT 
energy 

T-i 

radians  per  sec. 

Electric  energy  .  .  . 
Kinetic  energy...  . 

Electric  power.  .  .  . 

W 

W 

p 

IT 

EQ 

T* 

realP 

number 

joule;  watt-hour 
joule 

watt;  kilowatt 
1 

apparent  P 
wattless  P 

Electrochemi- 
cal. 

(  El'  mag.  system.) 
Ionic  charge.  .  . 

apparent  P 
Q 

coulomb  per  uni- 

Electrochemical 
equivalent  

Electric  deposition 

M 
M 
Q 
M 
T 

T'" 

valent  gram  ion 

gram     per     cou- 
lomb 

gram  per  second 

*  N  =  number  of  turns. 


t  0  =  temperature. 


24 


INTRODUCTION. 


PHYSICAL   QUANTITIES  AND   RELATIONS  (Continued}. 


Name. 

Sym- 
bol. 

Derivation. 

Dimensional 
Formula. 

Ratio  of 
Electro- 
static to 
Elec  'mag- 
netic Units. 

Electric. 

(Electrostatic  system.) 

E 

Resistance 

R,  r 

v—2 

~T 

Resistivity    or    specific 

1 

resistance  

P 

T(k~*) 

v-2 

r 

Reactance 

X,x 

x   

jj—  iffif  —  1\ 

v—2 

Magnetic  reactance.  .  .  . 

2nnL 

L-iTO-i) 

Capacity    reactance    or 

1  * 

1 

7-1         -1 

-2 

condensance 

\     c 

Impedance        • 

Z,  z 

\&+* 

*-> 

„-* 

Conductance.  .  .  . 

G  g 

7 

LT~l(k) 

^ 

E 

Conductivity  or  specific 

Q 

V2 

T 

STE/L 

Admittance  ... 

Y,y 

1 

LT-i(k) 

V2 

z 

Susceptance.  .  .  . 

B,  b 

•vA/2  f]2 

LT~l(k} 

V2 

Electromotive   force   or 

w 

potential  

E,  e  - 

L^M^T~^(k~^} 

v~  * 

~Q 

Difference    of    potential 

U,u 

ei-e2 

L^M^T~l(k~^) 

v~  l 

Electromotive   force    at 

i 

E 

111 

a  point.  . 

r   .  .  .  . 

Y 

lj      JVl    I      (K    * 

V 

Intensity  of  electric  field 

F 

Q 

L-Wr-^ 

fa 

ET 

Vector  potential  

L-%M^(k-%) 

v~i 

~J7 

I   i 

Q 

L$M$T-2(k$) 

V 

' 

T 

Current  density.  .  .  . 

I 

L-^M^T~2(k^) 

V 

S 

Quantity  of  'electricity, 

charge  

Q,  Q 

\/  L2F 

L%M^T-l(k^) 

V 

Surface  density  or  elec- 

I 

Q 

tric  displacement  

)    ' 

8 

L     M  T   l(k  ) 

V 

Capacity  

C,  c 

Q 

L 

V2 

E 

Electric    induction    ca- 
pacity    or     Dielectric 
constant  or  specific  in- 

u 

~L 

C  .  C 

number  (k) 

V* 

ductive  capacity  

L  '  L 

number 

1 

PHYSICAL    QUANTITIES    AND    RELATIONS.          25 


PHYSICAL   QUANTITIES  AND   RELATIONS  (Continued). 


Name. 

Sym- 
bol. 

Derivation. 

Dimensional 
Formula. 

Ratio  of 
Electro- 
static to 
El  ec  'mag- 
netic Units. 

Inductance  or  coefficient 
of     self  -  induction    or 
electrokinetic  inertia.  . 
Mutual  inductance     •  •  . 

},, 

xm 
2nn 

L-i!T2(jfc-i) 
L-iT2(k-i) 
T 

T 
T-i 

L*M*(fc-*) 
L*M* 

tr-2 

tr* 
1 

1 
1 

v-i 

r-i 

v-» 

v-i 

1 
1 

V 

V 

1 

L 
R 
1 

n 

Period  

Frequency     . 

n 

Electrokinetic    momen- 
tum   
Thermo-electric     height 
or  specific  heat  of  elec- 
tricity   
Coefficient  of  Peltier  ef- 
fect 

}•••• 

/Xinductance 
E 

e 

heat 
IT 
energy 
IT 
EQ 
W 
T 

Q 

M 
M 
Q 
M 
T 

QT 
Q 
T 

0 

Te(k*) 
L-%M*Te(k-*) 

L*MlT-l(k-l) 
L2MT~2 
L2MT~* 

LlM~*T-l(k*) 
L-%M*T(k-*) 
MT-i 

L2MT-2 
L2MT~* 

L*MT~3 
MI7"3 
MT~3 

Ditto 

Electric  energy  

W 
P 

Electric  power  

Electrochemical. 

(Electrostatic  system.) 
Ionic  charge.  .  .  . 

Electrochemical  equiva- 
lent   

Electric  deposition.  .  .  . 

Photometric.* 

Quantity  of  light 

Q 

0 

I 

e 
E 

Unit. 

lumen  hour 
lumen 

candle  ;  hef- 
ner 

candle    per 
sq.  cm 

lux 

Flux  of  light  

solid  angle 
7 
S 
9 

S 

Illumination  

*  The  dimensional  formulas  for  the  photometric  quantities  here  given 
have  been  deduced  by  the  author  on  the  basis  that  radiated  light  is  power, 
or  that  quantity  of  light  is  energy;  according  to  this,  the  formula  for  can- 
dle-power is  that  for  power  divided  by  that  for  a  solid  angle;  the  latter, 
having  the  dimensional  formula  1  in  the  C.  G.  S.  system,  is  a  "suppressed" 
factor  which  does  not  appear  in  the  formula. 


26 


INTRODUCTION. 


PHYSICAL  QUANTITIES  AND   RELATIONS  (Concluded). 


Name. 

Symbol. 

Derivation. 

Dimensional  Formulas.  t 

Dynamical 

Thermal 

Thermo- 
metric. 

»=H+M. 

Thermal.* 

Heat       .    .. 

H 

energy 

L2MT~2 

MB 

L30 

L2MT—* 

Rate    of    heat 
production... 

\ 

H 
T 

L2MT~* 

Temperature.  . 

ft 

n 

01 

0J 

T  2jr—  2 

Coefficient      of 
expansion.  .  . 

(V-V) 

0-1 

,-. 

0-1 

L-*r» 

V(0-0') 

Entropy.  .   . 

H 

e 

L2MT-20-1 

M 

L3 

M 

Latent  heat.  .  . 

H 
M 

L>I- 

0 

L3M-10 

VT-' 

Conductivity.  . 

HL 

TL2d 

LMT-0-1 

L-iMT-i 

L2T-1 

L-WT-i 

Emissivity     or 

ff 

immissivity.  . 

M  T—  30—  i 

L-2MT-1 

LT~  i 

L~2MT~^ 

TL2d 

Specific  heat. 

H 
H 

number 

number 

number 

number 

Capacity  

MXsp.  heat 

M 

M 

A/ 

M 

Mech  ani  c  a] 
equivalent..  . 

J 

mec.   energy 

number 

L2T   20-! 

M 

number 

heat-energy 

LT20 

*  For  the  names  and  definitions  of  heat-units  see  tables  of  units  of  Energy. 

t  The  dimensional  formulas  of  the  various  thermal  quantities  or  relations 
are  still  a  matter  of  some  conjecture,  excepting  only  that  of  quantity  of 
heat,  which  is  simply  energy,  and  that  of  the  rate  of  production  or  trans- 
mission of  heat  or  radiant  heat,  which  is  simply  power;  that  of  tempera- 
ture is  the  uncertain  factor.  For  this  reason  four  different  systems  are 
here  given,  based  on  four  different  fundamental  conceptions.  The  first  is 
based  on  the  dynamical  units,  that  is,  on  the  formula  of  energy  combined 
with  a  fourth  fundamental  unit  representing  temperature.  The  second  is 
based  on  the  thermal  units;  quantity  of  heat  then  is  mass  X  temperature  ; 
in  this  system  the  specific  heat  of  water  is  unity  by  definition  and  is 
therefore  suppressed  in  the  formula.  In  the  third  system  volume  is  sub- 
stituted for  mass  in  the  second.  In  the  fourth  system  the  author  has 


. 

eliminated  0  by  defining  temperature  as  energy  per  unit  of  mass,  that  is,  a» 
specific  mass  energy.  Most  of  the  data  in  columns  3,  4,  5,  and  6  have  been 
taken  from  the  Smithsonian  Physical  Tables  prepared  by  Prof.  Thomas 


Gray;  the  present  author  has  extended  them  and  supplied  some  omissionsg; 
the  new  matter  has  been  approved  by  Prof.  Gray. 

t  By  giving  6  the  dimensional  formula  L2T~2,  those  based  en  the  dynam- 
ical units  and  on  the  thermal  units  become  identical;  they  are  given  in 
the  last  column.  Or  by  making  it  L~  1MT~2,  those  based  011  the  dynam- 
ical and  the  thermometric  units  become  identical.  L2T~2  is  the  dimen- 
sional formula  of  energy  per  unit  of  mass,  and  L~1MT~2  is  that  of  pres- 
sure. Ruecker  suggests  giving  0  the  formula  L2MT~~2,  which  is  energy. 
No  great  importance  should  be  attached  to  attempts  to  give  temperature  a 
formula,  as  they  are  all  merely  speculative. 


TABLES 

OP 

CONVERSION     FACTORS. 


GENERAL  REMARKS. 

In  the  following  set  of  tables  every  unit  which  is  used  to  measure  quan- 
tities is  given  with  its  value  in  terms  of  the  other  units  of  its  kind.  Such 
numbers  are  variously  termed  values,  reduction  or  conversion  factors, 
equivalents,  relations,  ratios,  constants,  etc.;  they  give  the  relations  be- 
tween the  different  units  and  enable  one  to  reduce  each  one  (or  a  quantity 
measured  by  it)  to  the  other  (or  a  quantity  measured  by  it)  by  means  of  a 
single  multiplication.  The  table  includes  all  the  values  and  relations 
usually  given  in  books  under  the  title  of  "Weights  and  Measures,  "  although 
thase  form  only  a  very  small  part,  as  by  far  the  greater  number  have  never 
before  been  published  together  as  a  complete  set  of  values. 

The  term  "Weights  and  Measures"  has  not  been  used  in  connection  with 
these  tables,  partly  because  it  is  not  apparent  why  weights  are  not  also 
measures,  and  partly  because  the  present  tables  consist  largely  of  various 
measures  rarely  if  ever  given  in  books  under  the  title  of  weights  and  meas- 
ures. 

Owing  to  the  very  large  number  of  units  belonging  in  such  a  table, 
their  classification  and  arrangement  becomes  important  in  order  to  enable 
one  to  find  any  particular  unit  readily.  All  units  for  measuring  the  same 
quantity,  that  is,  all  units  of  length  for  instance,  or  all  units  of  volume,  etc., 
have  here  been  brought  together  in  one  group,  and  in  each  group  they  are 
arranged  in  the  order  of  their  size ;  all  the  different  values  of  each  unit  are 
also  arranged  in  order  of  size. 

To  give  the  value  of  each  unit  in  terms  of  each  of  the  others  would  have 
made  the  tables  many  times  as  long,  and  they  would  have  become  unnec- 
essarily cumbersome,  as  the  majority  of  the  values  are  never  used;  the 
number  of  values  or  reduction  factors  have  therefore  been  limited  to  those 
likely  to  occur  in  practice ;  if  the  others  are  ever  needed  they  can  readily  be 
found  from  these. 

To  avoid  unnecessary  repetition  of  values,  all  units  capable  of  being 
reduced  from  one  to  the  other  have  here  been  put  in  the  same  group.  Thus 
the  group  of  units  of  volumes  includes  both  cubical  units  such  as  cubic  feet, 
etc.,  as  well  as  capacity  measures,  such  as  gallons,  liters,  etc.  All  energy 
units  are  similarly  grouped  together,  whether  they  be  mechanical,  thermal, 
or  electrical.  In  a  few  cases,  however,  such  units  have  been  separated  into 
different  groups;  forces,  for  instance,  have  not  been  given  together  with 
weights,  although  they  are  interconvertible,  and  electrical  resistances  have 
been  given  separately  and  not  with  velocities,  although  in  the  C.  G.  S. 
electromagnetic  system  of  units  they  are  velocities. 

Sometimes  some  of  the  units  in  one  group  are  for  measuring  entirely  dif- 
ferent kinds  of  quantities,  and  even  have  entirely  different  dimensions,  yet 
their  equivalents  are  the'  same,  and  they  have  therefore  been  brought 
together  to  save  repetition;  for  instance,  pounds  per  square  foot  may  de- 
note either  a  pressure  or  the  weight  of  sheet  metaj;  in  either  case  the  equiv- 
alents in  other  units,  such  as  kilograms  per  square  meter,  will  be  the  same. 
Similarly,  grams  per  centimeter  may  denote  either  the  weights  of  a  wire  or 

27 


28 


TABLES   OF   CONVERSION    FACTORS. 


a  surface  tension,  yet  its  equivalents  in  other  units  are  the  same.  Power 
and  momentum,  or  energy  and  torque,  are  other  illustrations. 

In  the  case  of  lengths,  surfaces,  volumes,  and  weights  (masses)  there  are 
so  many  unusual,  special  trade,  obsolete,  or  foreign  units  the  values  or 
reduction  factors  of  which  are  not  often  needed,  that  they  have  been  grouped 
separately,  so  as  to  make  the  main  table  less  cumbersome  to  use. 

The  reciprocal  values  have  been  given  in  all  cases  in  which  it  was  thought 
they  were  likely  to  occur  in  practice,  thus  reducing  all  calculation  to  a  mere 
multiplication,  as  distinguished  from  a  long  division.  They  are  given  in 
their  proper  places,  and  one  should  therefore  first  look  for  the  units  that 
one  wants.  Thus  to  convert  meters  into  feet,  see  value  under  Meters,  and 
to  convert  feet  into  meters,  see  value  under  Feet. 

Special  attention  is  here  called  to  the  approximate  values  which  have 
been  given  in  nearly  every  case.  These  have  been  carefully  chosen  with  a 
view  to  reduce  the  calculation  to  the  smallest  possible,  generally  to  a  multi- 
plication by  one  digit  and  a  division  by  another  single  digit,  followed  by 
pointing  off  the  decimal.  They  are  believed  to  be  the  simplest  values  that 
exist.  The  accuracy  of  all  these  approximate  values  is  within  2%  and 
often  within  1%;  they  are  therefore  sufficient  for  most  calculations. 

The  symbols  or  abbreviations  which  are  given  are  either  those  in  common 
use  or  those  which  have  been  recommended  by  societies,  journals,  or  indi- 
viduals, and  in  a  few  unimportant  cases  where  none  existed  they  have  been 
supplied  by  the  author  in  conformity  with  the  others.  The  advantages 
and  desirability  of  a  uniform  and  universal  system  of  abbreviations  or  sym- 
bols are  too  evident  to  need  further  comment  here. 

This  whole  system  of  tables,  which  is  not  a  compilation  but  a  complete 
recalculation,  has  been  based  on  the  very  best  fundamental  values  that 
are  obtainable.  Their  numerical  values  are  given  in  their  respective  places 
or  in  the  introductory  notes;  they  are  printed  in  bold-faced  type.  When- 
ever legally  adopted  values  existed,  as  for  instance  for  the  relations  between 
the  metric  and  the  older  units,  they  have  been  used.  In  other  cases  the 
fundamental  values  have  been  obtained  from  the  best  existing  sources,  and 
wherever  possible  those  values  have  been  chosen  which  have  been  deter- 
mined upon  by  the  best  authorities  and  are  used  by  them.  The  chief  of 
these  authorities  was  the  National  Bureau  of  Standards,  from  which  the 
author  has  obtained  the  legal  values  and  all  the  other  standard  values  used 
or  recommended  by  it,  besides  much  valuable  assistance;  among  the  others 
were  the  International  Congresses,  the  Director  of  the  Nautical  Almanac, 
the  Coast  Survey  Department,  etc.  The  greatest  care  was  taken  in  obtain- 
ing all  these  fundamental  values,  and  it  is  believed  that  they  are  the  best 
which  exist  at  the  present  time.  They  are  limited  to  only  those  which  are 
absolutely  necessary,  as  was  explained  above  under  the  inter-relations  of 
units,  and  none  of  them  is  therefore  inconsistent  with  any  other,  nor  can  any 
derived  value  then  have  two  values  depending  upon  which  way  it  has  been 
calculated  from  the  fundamental  values;  all  the  values  together  form  a 
single,  uniform,  stable  system. 

The  fundamental  values  have  here  been  given  to  as  many  places  of  figures 
as  in  the  original  source.  The  derived  values  have  been  given  throughout 
to  six  significant  figures.  In  some  cases  the  fundamental  value  itself  may 
not  have  six  figures,  or  its  possible  error  may  not  warrant  so  many  places 
of  figures  in  the  derived  units;  but  in  such  a  completely  recalculated  set 
of  values  it  was  thought  best  to  retain  six  places  of  significant  figures 
throughout  in  all  the  derived  values,  as  the  correction  due  to  any  subse- 
quently adopted  more  accurate  fundamental  value  can  then  be  made  by 
mere  proportion  instead  of  by  a  complete  recalculation.  The  derived  values 
are  exactly  correct  for  the  particular  fundamental  values  used  here,  except- 
ing of  course  the  usual  slight  inaccuracy  (±1)  of  the  last  right-hand  digit 
in  the  numbers  and  in  the  logarithms. 

The  calculations  have  all  been  made  with  the  greatest  care;  each  was 
checked  by  calculating  it  in  another  way  and  in  many  cases  the  values  were 
thus  checked  twice;  it  is  therefore  believed  that  there  are  no  errors.  There 
is,  of  course,  the  unavoidable  uncertainty  of  one  unit  in  the  sixth  place  of 
figures  or  in  the  seventh  place  of  the  logarithms,  although  in  most  of  these 
cases  the  next  place  was  also  calculated  in  order  to  insure  the  accuracy  of 
those  that  were  retained.  Many  of  the  values  in  the  first  few  groups  have 
been  carefully  checked  by  L.  A.  Fischer,  Assistant  Physicist  of  the  National 
Bureau  of  Standards,  and  may  therefore  he  Assumed  to  be  those  used  by 


LENGTHS.  29 

that  Bureau.  The  author  takes  this  opportunity  to  acknowledge  his  appre- 
ciation of  Mr.  Fischer's  very  valuable  revision  of  those  values. 

The  obsolete  and  foreign  units  were  compiled  from  various  sources.  They 
therefore  involve  any  inaccuracies  or  inconsistencies  that  may  exist  in  the 
original  sources;  but  when  the  inconsistency  was  very  great,  care  was  taken 
to  find  which  value  was  the  correct  one. 

In  the  cases  in  which  units  are  sometimes  used  incorrectly  or  ambigu- 
ously, they  have  here  also  been  placed  where  they  might  be  looked  for,  and 
are  there  accompanied  by  a  reference  to  the  place  where  they  properly  be- 
long and  where  their  values  are  given. 

A  comparison  of  the  values  in  these  tables  with  those  in  other  books  will 
show  that  few  of  the  latter  have  been  based  on  the  legal  standards  of  this 
country.  Moreover,  the  older  published  values  will  often  be  found  to  be 
inconsistent  with  each  other,  as  they  seem  to  have  been  compiled  instead  of 
being  recalculated  throughout  from  the  same  fundamental  values,  as  was 
done  here. 

Special  attention  is  called  to  the  compound  uuits,  most  of  which  are 
seldom,  if  ever,  found  in  other  books;  as  most  of  these  save  from  two  to  four 
separate  calculations,  they  will  often  be  found  useful. 

LENGTHS. 

The  fundamental  standard  of  length  of  the  United  States  is  the  Inter- 
national Meter,  a  bar  made  of  an  alloy  of  90  percent  platinum  and  10  per- 
cent iridium  and  preserved  at  the  International  Bureau  of  Weights  and 
Measures,  near  Paris.  Copies  of  this  bar  are  possessed  by  each  of  the  twenty 
countries  contributory  to  the  support  of  the  International  Bureau  of  Weights 
and  Measures,  and  these  copies  are  known  as  National  Prototypes.  The 
United  States  owns  two  of  these  bars,  whose  values  in  terms  of  the  Inter- 
national Meter  are  known  with  the  greatest  accuracy.  One  of  these  bars, 
No.  21,  is  used  as  a  working  standard,  and  the  other,  No.  27,  is  kept  under 
seal  and  only  used  to  check  the  value  of  No.  21.  One  of  the  objects  of  the 
maintenance  of  the  International  Bureau  is  to  provide  for  the  recomparison 
at  regular  intervals  of  the  various  National  Prototypes  with  the  International 
Meter,  thus  insuring  the  use  of  the  same  standard  throughout  the  world. 

According  to  the  Act  of  Congress  of  July  28,  1866,  which  was  the  first 
general  legislation  in  the  United  States  upon  the  subject  of  fixing  the  stand- 
ards of  weights  and  measures,  the  relation  1  meter  ==39. 370  0  inches  was 
legalized,  and  it  is  the  only  legal  relation  which  exists  in  the  United  States 
concerning  the  meter.  This  value  is  the  one  used  by  the  National  Bureau 
of  Standards  in  Washington  and  is  in  general  use  in  this  country.  Since 
1893  the  Office  of  Standard  Weights  and  Measures  has  been  authorized  to 
derive  the  yard  from  the  meter  in  accordance  with  this  relation;  the  legal 
yard,  foot,  inch,  etc.,  are  therefore  derived  from  the  meter,  and  consequently 
are  fixed  and  definite  units.  The  relation  between  the  U.  S.  yard  and  the 
meter  is  therefore  no  longer  to  be  determined  by  measurement,  as  is  often 
supposed,  but  is  fixed  definitely  by  precise  definition.  There  is  no  legal 
authority  in  this  country  for  the  old  Kater  relation  of  1818,  namely, 
39.37079,  which  is  still  in  use,  notably  by  one  well-known  maker  of  gauges. 

In  Great  Britain  the  relation,  legalized  in  1896,  between  this  same  Inter- 
national Meter  and  the  British  Imperial  yard  of  36  inches  is  1  meter  = 
39.370  113  inches.  The  Kater  value,  39.370  79,  was  used  there  for  trade 
purposes  until  1896.  In  1867  the  Clark  value,  39.370  432,  was  recom- 
mended by  the  Warden  of  Standards,  London,  to  supersede  the  Kater 
value,  and  was  generally  used  in  scientific  work  in  Great  Britain  until  1896. 
There  exists  therefore  a  very  slight  difference  between  the  present  U.  S.  and 
British  legal  yard,  foot,  inch,  etc.  But  these  differences  are  only  about  3 
parts  in  one  million,  the  U.  S.  yard  being  the  longer;  it  is  therefore  abso- 
lutely negligible  except  in  the  most  refined  physical  measurements.  Unless 
otherwise  stated,  the  values  in  the  following  tables  are  based  on  the  U.  S. 
legal  relation. 

The  value  of  the  nautical  mile  in  meters  (1  853.25)  is  the  one  adopted 
many  years  ago  by  the  U.  S.  Coast  and  Geodetic  Survey,  and  has  not  been 
changed  since  its  first  adoption.  It  is  the,  length  of  one  minute  of  arc  of 
a  great  circle  of  a  true  sphere  whose  area  is  equal  to  that  of  the  earth.  The 
"Committee  Meter"  used  by  the  U.  S.  Coast  and  Geodetic  Survey  prior 
to  1889  is  equal  to  the  international  meter  of  39.370  0  inches  (U.  S.). 


30  LENGTHS 


LENGTHS,    Usual. 

**  Accepted  by  the  National  Bureau  of  Standards. 

*  Checked  by  L.  A.  Fischer,  Asst.  Phys.  National  Bureau  of  Standards. 

Aprx.  means  within  2%. 

Logarithm 

1  mil  =  0.025  400  05*  millimeter.    Aprx.  *4o 2-404  8348 

"   =  0.001  inch 3-000  0000 

I  millimeter  [mm]  =         39.370  0*  mils.    Aprx.  40 1-595  1654 

=  0.039  370  0*  inch.    Aprx.  V25 2-5951654 

0.001    meter 3-000  0000 

1  centimeter  [cm]  =      0.393  700*  inch.    Aprx.  Vio 1-595  1654 

=  0.032  808  3*  foot.    Aprx.  Vao 2-515  9842 

=  0.01    meter 2-000  0000 

1  inch [in]  =  1  000.    mils 3-000  0000 

=       25.400  05*  millimeters.    Aprx.  \i  X  100 1.404  8346 

=       2.540  005*  centimeters.    Aprx.  1% 0-404  8346 

=   0.083  333  3*  foot  or  Vi2 2-920  8188 

=   0.027  777  8*  yard  or  y36.    Aprx.  11A  + 100 2-443  6975 

=  0.025  400  05*  meter.    Aprx.  *40 2-404  8346 

1  decimeter  [dm]  =  10.  centimeters 1-000  0000 


3.937  00*  inches.  Aprx.  4. 
=  0.328  083*  foot.  Aprx.  V8.  -  - 
=  0.109  361*  yard.  Aprx.  V9.  . 


0.1    meter... 


1  foot  [ft]  (Brit.)=  0.999  997  1*  foot  (U.  S.).    Aprx.  1.. 
0.304  800*  meter.    Aprx.  8A0.  .  .  . 


-595  1654 
-515  9842 
-038  8629 
-000  0000 
•999  9988 
-484  014C 

1  foot  [ft]  (U.  S.)  =            304.801*  millimeters.    Aprx.  300 2-484  0158 

30.480  1*  centimeters.    Aprx.  30 1-484  0158 

12.  inches 1-079  1812 

3.048  01*  decimeters.    Aprx.  3 0-484  0158 

=      1.000  002  9*  feet  (Brit.).    Aprx.  1 ' 

0.333  333    yard  or  V3 

0.304  801*  meter.    Aprx.  %o 

=  0.000  304  801*  kilometer.    Aprx.  3  -*-  10  000 . 
=  0.000  189  394*  mile.    Aprx.  19-4-100  000.  . .  . 

1  yard  [yd]  (Brit.)  =  0.999  997  1*  yard  (U.  S.).    Aprx.  1 

=  0.914  399  2*  meter.    Aprx.  »/io  or  i°/n 

1  yard  [yd]  (U.  S.)  =  91.440  2*    centimeters.    Aprx.  90 

=                     36.     inches 1-556  3025 

=                        3.     feet 0-477  1213 

=      1.0000029*    yards(Brit.).    Aprx.  1.  ...  0-000  0012 

=         0.914  402**  meter.    Aprx.  %o  or  i%i .  -  1-961  1371 

=  0.000914402*    kilomt'r.    Aprx.  Hi  •*•  100.  4-961  1371 

=  0.000568182*    mile.    Aprx.  #  -r-  1  000 4-7544873 

1  meter  [m]=                 1  000.      millimeters 3-000  0000 

=                     100.      centimeters 2-000  0000 

=   39.370113*     inches  (Brit.).    Aprx.  40 1-5951668 

=  39.370000**  inches  (U.S.).    Aprx.  40 1. 595  1654 

=                      10.      decimeters 1-000  0000 

=           3.280  83**   feet.    Aprx.  1% 0-515  9842 

=           1.093  61**  yards.    Aprx.  1^0 Q-038  8629 

=         0.546  806*     fathom.    Aprx.  ^20 1-737  8329 

=                0.001       kilometer 3-000  0000 

=  0.000  621  370*     mile.    Aprx.  %-s- 1  000 4-793  3503 

1  kilometer  [km]  =    3  280.83*  feet.    Aprx. •%  X  10  000 3-5159842 

•    =    1  093.61*  yards.    Aprx.  1100 3-0388629 

=          1  000.  meters 3-000  0000 

=  0.621  370*  mile.    Aprx.  % 1-793  3503 

=  0.539  611    knot  (Brit.).    Aprx.  %i 1-732  0806 

44                 =0.539  593    nautical  mile  (U.  S.)  Aprx.  6/ii.  .  T.-732  0660 


LENGTHS.  31 

1  mile  [ml]  ==  same  as  statute  mile  or  land  mile. 

=         5  280.*  feet.    Aprx.  5  300 3.722  6339 

=         1  760.*  yards.    Aprx.  %  X 1  000 3-2455127 

=    1  609.35*  meters.    Aprx.  1  600 3-206  6497 

=    1.60935*  kilometers.    Aprx.add%o 0-2066497 

=  0.868  421     knot  (Brit.).    Aprx.  subtract  Vs 1-938  7303 

=  0.868  392     nautical  mile  (U.  S.).    Aprx.  subt.  Vs.  .  1-938  7157 
1  knot  or  nautical  mile  (Brit.): 

=       6  O8O.  feet.    Aprx.  6  000 3-783  9036 

=   2  026.67  yards.    Aprx.  2  000 3-306  7823 

=    1.853  19  kilometers.    Aprx.  H6 0-267  9194 

=    1.151  52  miles.    Aprx.  add  ty 0-061  2697 

=  0.999  966  nautical  mile  or  knot  (U.S.).    Aprx.  1 1-999  9854 

1  nautical  mile  or  knot  (U.S.)  same  as  geographical  or  sea  mile: 

=    6  080.20  feet.    Aprx.  6  000 3-783  9182 

=    2  026.73  yards.    Aprx.  2  000 3-306  7969 

=  1  853.35  meters.    Aprx.  H'e X  1  000 3-267  9340 

=    1.853  25  kilometers.    Aprx.  1% 0-267  9340 

=    1.151  55  miles.    Aprx.  add  Vr 0-061  2843 

=  1.000034]  knots  (Brit.).    Aprx.  1 0-0000146 

«=  1  minute  of  earth's  circumference 0-000  0000 


LENGTHS  (continued).     Unusual,  Special  Trade,  or  Obsolete. 

1  Angstroem  unit  (spectroscopy)  =  0.1  milli-micron,  or  micro-milli- 
meter =  0.000  1  micron  =  0.000  003  937  00  mil  =  0.000  000  1  millimeter. 
1  mil  =  254  000.5  Angstroem  units. 

1  milli-micron  [^^]  (spectroscopy)or  micro-millimeter  (microscopy) 
=  10  Angstroem  units  =  0.001  micron  =  0.000  039  370  0  mil  =  0.000  001  milli- 
meter. 1  mil  =  25  400.1  milli-microns.  The  term  micro-millimeter  is  also 
used,  though  incorrectly,  in  biology  for  0.001  millimeter,  which  length  is 
wiore  properly  called  a  micron  or  micro-meter. 

1  wave  length  of  blue  light  is  of  the  order  of  about  5000  Angstroem 
Units  or  500  milli-microns  or  0.5  micron  or  0.02  mil.  For  accurate  values 
see  below  under  meter. 

1  micron  or  microne  or  micro-meter  [/*]  (spectroscopy  and  micros- 
copy) =10  000  Angstroem  units  =  1000  milli-microns  or  micro-milli- 
meters =0.039  370  0  mil  (aprx.  %5)  =  0.001  millimeter.  1  mil  =  25.400  1 
microns. 

1  terze  (Brit.)  =  Vi2  second  =  Vi44  Hne  =  :H728  inch. 
1  second  (Brit. )  =  12.  terzes  =  Vi2  1^16  =  ^44  inch. 
1  point  =  0.008  inch. 

1  point  (typography)  =  %  line  =  Vr2  inch. 
1  line  (U.  S.)  =  Vi2  inch. 

1  line  (Brit. )  =  144.  terzes=  12.  seconds=Ha  inch.    Also  given  as  Ho  inch. 
1  hairsbreadth  =  M  line  =  Ms  inch. 
1  barleycorn  =  J^  inch. 
1  nail  [na]  (cloth)  =  2M  inches  =  M  span. 
1  palm  =  3.  inches. 
1  hand  =  4.  inches. 

1  link  [li]  (surveyor's)  =  7. 920  0  inches  =  0.201  17  meter. 
link  [li]  (engineer's)  =  12.  inches  =  l.  foot  =  0.304  80  meter, 
span  =  9.  inches  =  4.  nails  =1.  quarter  =  ^4  yard. 
quarter  [qr]  (cloth)  =  9.  inches  =  4.  nails  =  l.  span  =  ^4  yard. 
cubit  =18.  inches=iy2  feet. 
cubit  (in  Bible)  =  21.8  inches. 
vara  (California;   legal)  =  33. 372  inches. 
pace  =  3.  feet. 
military  pace  =  3.  feet. 
common  pace  =  2^  feet. 

meter  used  by  Pratt   &  Whitney  Co.  =  39.370  79  inches -1.000  020 
meters  (int.). 


32 


LENGTHS. 


1  meter  =  1  553  163.5  wave  lengths  of  red  light. t 
=  1966249.7  "  "  green  "  f 
=  2083  372.1  ' blue  "  t 

1  Committee    Meter    (of    U.  S.  Coast    Survey)  =  39. 370  0    inches  =1 
meter  (int.;. 

1  ell  [E,  e]  (cloth;  Brit.)  =  45.  inches -1,143  0  meters. 

1  fathom  (U.  S.)  =  6.  feet  =  1.828  8  (aprx.  %)  meters. 

1  fathom  (Brit.)  =  6.080  feet  =  1.853  2  (aprx.  1%)  meters  =  Hooo  nauti- 
cal mile  (Brit.). 

1  rod  [r],  pole  [p],  or  perch  [p]  (surveyor's)  =  5^  yards  =  5.029  2  meters. 

1  decameter  or  dekameter  [dkm]=10.  meters. 

1  chain  [ch]  (Gunter's  or  surveyor's)  =  100.  links  (surveyor's)  =  66.  feet  = 
20.117  meters  =  4.  rods,  poles,  or  perches  =  Vio  furlong^/ko  mile. 

1  chain    [ch]     (engineer's)  =  100.    links  (engineer's)  =  100.  feet  =  30.480 
meters. 

1  chain  [ch]  (Philadelphia  standard)  =  100*4  feet  =  30.556  meters. 
bolt  (cloth)  =  40.  yards. 


Ibo       

1  hectometer  = 


100.  meters. 


1  furlong  [fur]  =  660.  feet  =  20 1.1 7  meters  =  40.  rods,  poles,  or  perches  = 
10.  chains  (Gunter's  or  surveyor's)  =  V8  mile. 

1  cable  or  cable's  length  (Brit,  navy)  =  608.  feet  (sometimes  stated  as 
608.6  feet)  =  Vio  nautical  mile  (Brit.). 

1  cable's  length  (U.S. Navy)  =  720. feet  =  219.457  meters  =  120. fathoms 
(U.  S.). 

1  car-mile,  see  under  units  of  Energy. 
1  knot  (telegraph;  Brit.)  =  2  029.  yards=l  855.32  meters. 
1  geographical  mile,  sometimes  used  for  nautical  mile;  see  also  under 
international  geographical  mile. 

1  international  nautical  or  sea  mile  =  6  076.10  feet  =  1  852.  meters  = 
%o  of  1°  of  meridian. 

1°  of  latitude  at  equator  =  60.  (aprx.)  nautical  miles. 

=  68.70  miles  (statute), 
lat.  20°  =  68.78 
40°  =  69.00 
60°  =  69.23 
80°  =  69.39 
90°  =  69.41 
1°  of  longitude  at  equator  =  60.  (aprx.)  nautical  miles. 


=  69.16  m 


les  (statute). 


"    lat.  20°  =  65.02 
"    "      40°  =  53.05 
"    ••      60°  =  34.67 
••    ••      80°  =  12.05 
1  legua  (California;   legal)  =  2.633  5  miles  =  5  000.  varas. 
1  league  (U.  S.)  =  4.828  05  kilometers  =  3.  miles  (statute);  also  given  ad 
3.  nautical  miles. 

1  international  geographical  mile  =  24  350. 3  feet  =  7  422.  meters  = 
4.611  80  miles  (statute)  =  4.  (aprx.)  nautical  miles  =  ii5  of  1°  at  equator. 
The  term  geographical  mile  is  sometimes  used  also  for  nautical  mile. 
1  myriameter  or  miriameter=10.  kilometers. 

1  mean  diameter  of  the  earth  J  (astronomy )  =  12  742.0  kilometers  = 
7  917.5  miles. 

1  mean  diameter    of     earth's    orhit§     (astronomy)  =  149  340  870  ± 
9G  101  kilometers  =  92  796  950  ±59  715  miles. 


t  Michelson;    cadmium  light  waves  for  air  at  15°  C.  and  a  pressure  of 
^0  mm.  mercury. 

t  Based  on  the  accepted  value  of  a  nautical  mile  as  defined  above. 
§  Harkness. 


LENGTHS.  33 


LENGTHS  (concluded).    Foreign. 

These  are  mostly  obsolete,  as  the  metric  system  is  now  used  in  most 
foreign  countries.  The  British  measures  are  included  among  the  U.  S. 
measures,  being  very  nearly,  and  sometimes  quite  the  same.  The  trans- 
lated terms  are  merely  synonymous,  and  not  the  exact  equivalents. 

Germany.  Prussia.  Legal  May  16,  1816.  1  Fuss  [']  (foot)  =  12  Zoll  ["] 
(inches)  of  12  Linicn  ['"]  (lines).  1  Fuss  (also  called  "  rheinlaendischer 
Fuss,"  that  is,  Rhineland  foot)  =  0.313  853  5  meter.  Road  measure:  1 
Meile  (mile) -2  000  Ruthen  (rods)  of  12  Fuss  (feet);  1  Meile  =  7.5325 
kilometers;  1  Ruthe  =  3. 766  25  meters.  From  Jan.  1, 1872, to  Jan.  1,  1874: 
1  deutsche  Meile  (German  mile)  =  7. 500  kilometers.  Trade  measure:  1 
Elle(yard)  =  2H  Fuss  (feet)  =  25^  Zoll  (inch)  =  0.666  939  meter.  1  Lach- 
ter  =  80  Zoll  (inches)  =  2.092  36  meters. 

Bavaria.  1  Fuss  (foot)  =  12  Zoll  (inches)  of  12  Linien  (lines).  More 
rarely  1  Fuss=  10  Zoll  of  10  Linien.  1  Fuss  =  0.291  859  meter. 

Saxony.     1  Fuss  =  0.283  19  meter;   subdivisions  like  in  Bavaria. 

WiLertemberg.     1  Fuss  =  0.286  49  meter;   subdivisions  like  in  Baden. 

Baden.  1  Fuss  (foot)  =  10  Zoll  (inches)  of  10  Linien  (lines).  1  Fuss  = 
0.3  meter. 

Hanover.     1  Fuss  =  0.292  1  meter. 

The  following  values  of  various  German  feet  in  inches  are  given  in  Nys- 
trom's Mechanics:  Bavaria,  11.42;  Berlin.  12.19;  Bremen,  11.38;  Dres- 
den, 11.14;  Hamburg,  11.29;  Hanover,  11.45;  Leipsic,  11.11;  Prussia, 
12.36;  Rhineland,  12.35;  Strasburg,  11.39.  Also  the  following  road  meas- 
ures: Germany,  mile,  long,  10126  yards;  Hamburg  mile,  8244  yards; 
Hanover  mile,  11  559  yards;  Prussia  mile,  8  468  yards. 

France.  "Old  measures"  (systeme  ancien)  used  prior  to  1812:  1  toise 
(fathom)  =  6  pieds  (du  roi)  (feet)  of  12  pouces  (inches)  of  12  lignes  (lines) 
of  12  points.  In  geodesy:  1  pied=10  pouces  of  10  lignes  of  10  points. 
1  toise  =  1.949  037  meters.  Road  measure:  1  lieue  (league)  =  2  283  toises  = 
4449.65  meters;  1  lieue  marine  =  2854  toises  =  5  562.55  meters;  1  lieue 
moyenne  =  2  534  toises  =  4  938.86  meters.  Field  measure:  1  perche 
(perch)  =  18  or  22  pieds.  For  depths  of  the  sea:  1  brasse  =  5  pieds.  Trade: 
1  aune  (yard)  de  Paris=  1.188  45  meters.  According  to  Nystrom's  Me- 
chanics: 1  pied  du  roi=  12.79  inches;  1  league  (lieue)  marine  =  6  075  yards; 
1  league,  common  =  4  861  yards;  1  league,  post,  =  4  264  yards. 

"Usual"  measures  (systSme  usuel)  used  from  1812  to  1840:  1  toise 
(fathom)  =  2  meters;  1  pied  (foot)  =  M  meter;  1  aune  (yard)  =  1.2  meters. 
For  subdivisions  into  other  units  see  above  under  "old  measures."  1  noeud 
=  1  knot  or  nautical  mile. 

Austria.  1  Ruthe  (rod)  =  10  Fuss  (feet)  of  12  Zoll  (inches)  of  12  Linien 
(lines).  1  Fuss  =  0.316  10  meter.  1  Meile  (mile)  =  7.586  kilometers  = 
4  000  Klafter  of  6  Fuss.  1  Elle  (yard)  =  2.46  Fuss.  According  to  Nys- 
trom's Mechanics:  1  Vienna  foot  =  12. 45  inches. 

Sweden.  1  meile  (mile)  =  6  000  famn  of  3  alnar  of  2  fot  (foot)  of  10  turn 
(inches)  (or  12  vorktum)  of  10  linier  (lines).  1  meile  =  10.688  4  kilometers. 
1  fot  =  0.296  901  meter.  1  ruthe  (rod)  =  16  fot.  1  corde=10  stangen 
of  10  fot.  According  to  Nystrom's  Mechanics:  1  Swedish  foot  =  11.69 
inches;  1  Swedish  mile  =11  700.  yards. 

Russia.  1  werst  =  500  saschehn  of  3  arschin  of  4  tschetwert  of  4  wer- 
schock.  1  werst  =  1.066  8  kilometers  =  3  500  feet  =  0.662  88  mile.  1  sa- 
schehn =2. 133  6  meters  =  7.0  feet  (U.S.).  1  arschin  =  7 1.1 2  centimeters  = 
28.0  inches.  1  tschetwert  =  17.780  centimeters  =  7.0  inches.  1  werschock  = 
4.445  centimeters  =1.75  inches.  The  British  foot  is  also  used;  1  foot  = 
12  inches  of  12  lines.  According  to  Nystrom's  Mechanics:  1  Russian 
foot=13.75  inches;  1  Moscow  foot  =  13.17  inches;  1  Riga  foot  =  10.79 
inches;  1  Warsaw  foot  =  14.03  inches;  1  verst  =  l  167.  yards. 

Switzerland.  1  Fuss  (foot)  =  10  Zoll  (inches)  of  10  Linien  (lines). 
1  Fuss  =  0.3  meter.  According  to  Nystrom's  Mechanics:  1  Geneva  foot  = 
19.20  inches;  1  Zurich  foot  =  11.81  inches;  1  Swiss  mile  =  9  153  yards. 

Holland.  1  Ruthe  (rod)  =  12  Fuss  (feet).  1  Ruthe  =  3.767  36  meters; 
1  Fuss  =  0.313  947  meter.  According  to  Nystrom's  Mechanics:  1  Amster- 
dam foot  =  11. 14  inches;  1  Utrecht  foot  =  10.74  inches;  1  Flanders  mile  = 


34 


LENGTHS. 


6869  yards;  1  Holland  mile  =  6  395  yards;  1  Netherlands  mile  =  1093 
yards. 

Spain.  1  vara  (yard)  =  32.874  8  inches  =  0.835  022  meter.  According 
to  Nystrom's  Mechanics:  1  Spanish  foot  =  11. 03  inches;  1  toesas  =  66.72 
inches;  1  palmo  =  8.64  inches;  1  Spanish  common  league  =  7  416  yards. 

Italy.  According  to  Nystrom's  Mechanics:  1  Florence  braccio  =  21.69 
inches;  1  Genoa  palmo  =  9. 72  inches;  1  Malta  foot  =  11. 17  inches;  1  Naples 
palmo  =  10.38  inches;  1  Rome  foot  =  11. 60  inches;  1  Sardinia  palmo  = 
9.78  inches;  1  Sicily  palmo  =  9.53  inches;  1  Turin  foot  =  12. 72  inches; 
1  Venice  foot  =  13.40  inches;  1  Rome  mile  =  2  025.  yards. 

Japan.  Long  measure:  1  ri  =  36  cho  of  60  ken  of  6  shaku  of  10  sun  of 
10  bu.  1  jo  =  10  shaku.  1  ri  =  3.93  kilometers  =  2. 44  miles.  1  kilometer  = 
0.255  ri.  1  mile  =  0.410  ri.  For  cloth  measure:  1  jo  =  10  shaku  of  10  sun 
of  10  bu;  the  unit  is  the  jo;  in  this  measure  the  units  are  ^longer  than  those 
of  the  same  name  in  the  long  measure. 

Miscellaneous  (from  Nystrom's  Mechanics):  Antwerp  foot  =11. 24 
inches;  Brussels  foot  =  11.45  inches.  Denmark  mile  =  8  244  yards;  Copen- 
hagen foot  =12. 35  inches.  Portugal  league  =  6  760  yards;  Lisbon  foot  = 


12.96  inches;  Lisbon  palmo  =  8.64  inches.  Ireland  mile  =  3038  yards; 
Scotland  mile  =1984  yards.  Hungary  mile  =  9  113  yards.  Bohemia 
mile  =  10  137  yards.  Poland  mile,  long,  =8101  yards.  Turkey  berri  = 


1826  yards.  Persia  parasang  =  6  086  yards;  Persia  arish  =  38.27  inches. 
Arabia  mile  =  2  148  yards.  China  li  =  629  yards;  mathematic  foot=13.12 
inches;  builder's  foot  =  12. 71  inches;  tradesman's  foot  =  13. 32  inches; 
surveyor's  foot  =  12. 58  inches. 

Ancient.  Biblical:  1  digit  =  0.912  inch;  1  palm  =  4  digits  =  3. 648 
inches;  1  span  =  3  palms  =10.94  inches;  1  cubit  =  2  spans  =  21. 888  inches; 
1  fathom  =  3.46  cubits  =  7.296  feet.  Egyptian:  1  finger  =  0.737  4  inch; 
1  nahud  cubit  =  1.476  feet;  1  royal  cubit  =  1.722  feet.  Grecian:  1  digit  = 
0.754  inch;  1  pous=16  digits  =  1.007  3  feet;  1  cubit  =  1.133  feet;  1 
stadium  =  604. 375  feet;  1  mile  =  8  stadiums  =  4  835.  feet.  Hebrew:  1 
cubit  =  1.822  feet;  1  Sabbath  day's  journey  =  3  648.  feet;  1  mile  =  4  000 
cubits  =  7296  feet;  1  day's  journey  =  33. 164  miles;  1  sacred  cubit  =  2.02 
feet.  Roman:  1  digit=0.725  7  inch;  1  uncia  (inch)  =  0.967  inch;  1  pes 
(foot)  =  12  uncias=11.60  inches;  1  cubit  =  24  digits  =  1.45  feet;  1  passus  = 
3. 33  cubits  =  4.835  feet;  1  millarium  (mile)  =  4  842  feet.  Arabian:  1  foot  = 
1.095  feet.  Babylonian:  1  foot  =  1.14  feet. 

Lengths  in  which  Overhead  Telegraph  Lines  Are  or  Were  Ex- 
pressed in  Different  Countries.  (From  Munro  and  Jamieson's  Pocket 
Book.)  The  lengths  here  given  are  in  English  or  statute  miles  of  5  280  feet. 
Arabia,  mile  1.2204.  Austria,  mile  5.7534.  Bohemia,  mile  5.7596. 
Brabant,  league  3.452  2.  Burgundy,  league  3.516  6.  China,  li  0.359  1. 
Denmark,  mile  4.684  1.  Flanders,  league  3.90.  Hamburg,  mile  4.684  1. 
Hanover,  mile  6.567  6.  Hesse,  mile  5.992  6.  Holland,  mile  4.602  8.  Hun- 
gary, mile  5.1778.  Italy,  mile  1.1505.  Lithuania,  mile  5.5573.  -Nor- 
way, mile  7.018  3.  Oldenburg,  mile  6.147  7.  Poland,  long  mile  4.602  8: 
short  mile  3.4517.  Portugal,  league  3.8409.  Prussia,  mile  4.8068; 
Rome,  mile  0.925  0.  Russia,  verst  0.663  0.  Saxony,  mile  5.627  8.  Silesia, 
mile  4.0244.  Spain,  common  legua  of  8000.  varas,  4.2136;  legal  legua 
of  5  000  veras,  2.753  4.  Swabia,  mile  5.633  5.  Sweden,  mile  6.647  7. 
Switzerland,  mile  5.200  5.  Turkey,  berri  1.037  5.  Tuscany,  mile  1.027  2. 
Westphalia,  mile  6.903  9. 


LENGTHS. 


35 


Inches  in  Fractions,  Decimals,  Millimeters,  and  Feet. 

For  every  64th  of  an  inch  up  to  1  inch ;  for  every  32d  of  an  inch  up  to  6 
inches;  for  every  16th  of  an  inch  up  to  12  inches;  for  every  inch  ap  to  10 
feet.  The  equivalents  of  other  intermediate  values,  or  of  values  beyond 
the  table,  may  be  found  by  adding  together  two  or  more  values  from  the 
table;  thus  for  11%2  inches  add  that  for  11  inches  to  that  for  %2- 


Milli- 
meters. 

Inches. 

Feet. 

Milli- 
meters. 

Inches. 

Feet. 

0.396  876 

H4 

0.015  625 

0.001  302 

19.446  9 

4%4 

0.765  625 

0.063  80 

0.793  752 

J-152 

0.031  250 

0.002  604 

19.843  8 

25/82 

0.781  250 

0.065  10 

1.190  63 

%4 

0.046  875 

0.003  906 

20.240  7 

% 

0.796  875 

0.066  41 

1.587  50 

%6 

0.062  500 

0.005  208 

20.637  5 

13/16 

0.812  500 

0.067  71 

1.984  38 

%4 

0.078  125 

0.006  510 

21.034  4 

5%4 

0.828  125 

0.069  01 

2.381  25 

%2 

0.093  750 

0.007  813 

21.431  3 

27/32 

0.843  750 

0.070  31 

2.778  13 

%* 

0.109  375 

0.009  115 

21.828  2 

5%4 

0.859  375 

0.071  61 

3.17501 

y% 

0.125  000 

0.010  42 

22.225  0 

H 

0.875  000 

0.072  92 

3.571  88 

%4 

0.140  625 

0.011  72 

22.621  9 

57/64 

0.890  625 

0.074  22 

3.968  76 

%2 

0.156  250 

0.01302 

23.0188 

2%2 

0.906  250 

0.075  52 

4.365  63 

1M* 

0.171  875 

0.014  32 

23.415  7 

5%4 

0.921  875 

0.076  82 

4.762  51 

fc« 

0.187  500 

0.015  63 

23.812  5 

15/16 

0.937  500 

0.078  13 

5.159  39 

18/64 

0.203  125 

0.016  93 

24,209  4 

% 

0.953  125 

0.079  43 

5.556  26 

%J 

0.218  750 

0.018  23 

24.606  3 

% 

0.968  750 

0.080  73 

5.953  14 

15/64 

0.234  375 

0.019  53 

25.003  2 

63/04 

0.984  375 

0.082  03 

6.350  01 

M 

0.250  000 

0.020  83 

25.400  1 

1.000000 

0.083  33 

6.746  89 

17/64 

0.265  625 

0.022  14 

26.1938 

IMa 

1.031  250 

0.085  94 

7.143  76 

%2 

0.281  250 

0.023  44 

26.987  6 

IVio 

1.062  500 

0.088  54 

7.540  64 

A%4 

0.296  875 

0.024  74 

27.781  3 

1%2 

1.093750 

0.091  15 

7  93752 

5Ae 

0.312  500 

0.026  04 

28.575  1 

1H 

1.125000 

0.093  75 

8  334  39 

2l/64 

0.328  125 

0.027  34 

29.368  8 

1%2 

1.156250 

0.096  35 

8  731  27 

% 

0.343  750 

0.028  65 

30.162  6 

1%0 

1.187500 

0.098  96 

9  128  14 

23/64 

0.359  375 

0.029  95 

30.956  3 

!7/32 

1.218750 

0.101  6 

9.52502 

ys 

0.375  000 

0.031  25 

31.750  1 

1'4 

1.250000 

0.1042 

9.921  89 

2%4 

0.390  625 

0.032  55 

32.543  8 

1%2 

1.281  250 

0.106  8 

10.3188 

13/32 

0.406  250 

0.033  85 

33.337  6 

!5/4e 

1.312  500 

0.109  4 

10.715  6 

2T/64 

0.421  875 

0.035  16 

34.131  3 

1% 

1.343  750 

0.1120 

11.1125 

7/4e 

0.437  500 

0.036  46 

34.925  1 

w 

1.375000 

0.1146 

11.5094 

29/64 

0.453  125 

0.037  76 

35.7188 

l18/32 

1.406250 

0.1172 

11.906  3 

1B/32 

0.468  750 

0.039  06 

36.512  6 

IVie 

1.437  500 

0.1198 

12.303  1 

% 

0.484  375 

0.040  36 

37.306  3 

!15/32 

1.468750 

0.1224 

12.700  0 

H 

0.500  000 

0.041  67 

38.100  1 

VA 

1.500000 

0.1250 

13.096  9 

33/64 

0.515625 

0.042  97 

38.893  8 

!17/32 

1.531  250 

0.1276 

13.493  8 

17/32 

0.531  250 

0.044  27 

39.687  6 

1%6 

1.562  500 

0.1302 

13.890  7 

8%4 

0.546  875 

0.045  57 

40.481  3 

!19/32 

1.593750 

0.1328 

14.287  5 

ttS 

0.562  500 

0.046  88 

41.275  1 

VA 

1.625000 

0.1354 

14.684  4 

37/64 

0.578  125 

0.048  18 

42.068  8 

1% 

1.656250 

0.1380 

15.081  3 

!%2 

0.593  750 

0.049  48 

42.862  6 

1^6 

1.687500 

0.140  6 

15.478  2 

3%4 

0.609  375 

0.050  78 

43.656  3 

12%2 

1.718750 

0.1432 

15.875  0 

X 

0.625  000 

0.052  08 

44.450  1 

IX 

1.750  000 

0.1458 

16.271  9 

4y&4 

0.640  625 

0.053  39 

45.243  8 

125/S2 

1.781  250 

0.1484 

16.668  8 

% 

0.656  250 

0.054  69 

46.037  6 

113/16 

1.812  500 

0.151  0 

17.065  7 

4%4 

0.671  875 

0.055  99 

46.831  3 

!27/32 

1.843  750 

0.1536 

17.462  5 

iVlG 

0.687  500 

0.057  29 

47.625  1 

I-H 

1.875000 

0.156  3 

17.859  4 

45/64 

0.703  125 

0.058  59 

48.418  8 

12%2 

1.906250 

0.1589 

18.256  3 

23/32 

0.718  750 

0.059  90 

49.212  6 

I15/ie 

1.937  500 

0.161  5 

18.653  2 

47/64 

0.734  375 

0.061  20 

50.006  3 

1% 

1.968750 

0.164  1 

19.050  0 

fe 

0.750  000 

0.062  50 

50.800  1 

2 

2.000  000 

0.1667 

36 


LENGTHS. 


Milli- 
meters. 

Inches. 

Feet. 

Milli- 
meters. 

Inches. 

Feet. 

51.593  9 

2^2 

2.031  250 

0.169  3 

96.043  9 

325/32 

3.781  250 

0.315  1 

52.387  6 

21/16 

2.062  500 

0.171  9 

96.837  7 

313/16 

3.812  500 

0.317  7 

53.181  4 

23/32 

2.093  750 

0.1745 

97.631  4 

327/32 

3.843  750 

0.320  3 

53.975  1 

2X8 

2.125  000 

0.177  1 

98.425  2 

3j/s 

3.875  000 

0.322  9 

54.768  9 

2%2 

2.156  250 

0.1797 

99.218  9 

329/32 

3.906  250 

0.325  5 

55.562  6 

2«/16 

2.187  500 

0.182  3 

100.013 

315/10 

3.937  500 

0.328  1 

56.356  4 

27/32 

2.218  750 

0.1849 

100.806 

3% 

3.968  750 

0.330  7 

57.150  1 

2M 

2.250  000 

0.187  5 

101.600 

4 

4.000  000 

0.333  3 

57.943  9 

2%2 

2.281  250 

0.190  1 

102.394 

4H2 

4.031  250 

0.335  9 

58.737  6 

2%6 

2.312  500 

0.192  7 

103.188 

4^6 

4.062  500 

0.338  5 

59.531  4 

2% 

2.343  750 

0.195  3 

103.981 

4%2 

4.093  750 

0.341  1 

60.325  1 

2H 

2.375  000 

0.197  9 

104.775 

4^ 

4.125000 

0.343  8 

61.118  9 

213/32 

2.406  250 

0.200  5 

105.569 

4%2 

4.156  250 

0.346  4 

61.9126 

27/16 

2.437  500 

0.203  1 

106.363 

43/lfl 

4.187  500 

0.349  0 

62.706  4 

215/32 

2.468  750 

0.205  7 

107.156 

47/32 

4.218  750 

0.351  6 

63.500  1 

2H 

2.500  000 

0.208  3 

107.950 

4M 

4.250  000 

0.354  2 

64.293  9 

2'  %2 

2.531  250 

0.2109 

108.744 

4%2 

4.281  250 

0.356  8 

65.087  6 

2*ia 

2.562  500 

0.213  5 

1  09-.  538 

4%6 

4.312  500 

0.359  4 

65.881  4 

219/32 

2.593  750 

0.216  1 

110.331 

4% 

4.343  750 

0.362  0 

66.675  1 

2^ 

2.625  000 

0.2188 

111.125 

4% 

4.375  000 

0.364  6 

67.468  9 

2% 

2.656  250 

0.221  4 

111.919 

4i%2 

4.406  250 

0.367  2 

68.262  6 

2Hle 

2.687  500 

0.224  0 

112.713 

47/16 

4.437  500 

0.369  8 

69.056  4 

22%2 

2.718  750 

0.226  6 

113.506 

415/32 

4.468  750 

0.372  4 

69.850  1 

234 

2.750  000 

0.229  2 

114.300 

4M 

4.500  000 

0.375  0 

70.643  9 

225/32 

2.781  250 

0.231  8 

115.094 

417/32 

4.531  250 

0.377  6 

71.4376 

213,16 

2.812  500 

0.234  4 

115.888 

4ft/16 

4.562  500 

0.380  2 

72.231  4 

22%2 

2.843  750 

0.237  0 

116.681 

419/32 

4.593  750 

0.382  8 

73.025  1 

2^ 

2.875  000 

0.239  6 

117.475 

4H 

4.625  000 

0.385  4 

73.818  9 

22%2 

2.906  250 

0.242  2 

118.269 

4% 

4.656  250 

0.388  0 

74.612  7 

215/16 

2.937  500 

0.244  8 

119.063 

4H/16 

4.687  500 

0.390  6 

75.406  4 

2% 

2.968  750 

0.247  4 

119.856 

423/32 

4.718  750 

0.393  2 

76.200  2 

3 

3.000  000 

0.250  0 

120.650 

4M 

4.750  000 

0.395  8 

76.993  9 

3M2 

3.031  250 

0.252  6 

121.444 

425/32 

4.781  250 

0.398  4 

77.787  7 

31/16 

3.062  500 

0.255  2 

122.238 

413/16 

4.812  500 

0.401  0 

78.581  4 

3%2 

3.093  750 

0.257  8 

123.031 

427/32 

4.843  750 

0.403  6 

79.375  2 

3H 

3.125  000 

0.260  4 

123.825 

4Ji 

4.875  000 

0.406  3 

80.168  9 

35/32 

3.156  250 

0.2630 

124.619 

429/32 

4.906  250 

0.408  9 

80.962  7 

33/16 

3.187  500 

0.265  6 

125.413 

415/i6 

4.937  500 

0.411  5 

81.756  4 

3V32 

3.218  750 

0.268  2 

126.207 

4% 

4.968  750 

0.414  1 

82.550  2 

3^ 

3.250  000 

0.270  8 

127.000 

5 

5.000  000 

0.416  7 

83.343  9 

3%2 

3.281  250 

0.2734 

127.794 

5^2 

5.031  250 

0.419  3 

84.137  7 

3&/16 

3.312  500 

0.276  0 

128.588 

5^6 

5.062  500 

0.421  9 

84.931  4 

3% 

3.343  750 

0.278  6 

129.382 

5%2 

5.093  750 

0.424  5 

85.725  2 

3^ 

3.375  000 

0.281  3 

130.175 

5H 

5.125000 

0.427  1 

86.518  9 

313/32 

3.406  250 

0.283  9 

130.969 

55/32 

5.156  250 

0.429  7 

87.312  7 

37/16 

3.437  500 

0.286  5 

131.763 

53/16 

5.187  500 

0.432  3 

88.106  4 

315/32 

3.468  750 

0.289  1 

132.557 

57/32 

5.218  750 

0.434  9 

88.900  2 

&A 

3.500  000 

0.291  7 

133.350 

5M 

5.250  000 

0.437  5 

89.693  9 

317/S2 

3.531  250 

0.294  3 

134.144 

5%2 

5.281  250 

0.440  1 

90.487  7 

3»/16 

3.562  500 

0.296  9 

134.938 

5&/16 

5.312  500 

0.442  7 

91.281  4 

319/32 

3.593  750 

0.299  5 

135.732 

5% 

5.343  750 

0.445  3 

92.075  2 

3^8 

3.625  000 

0.302  1 

136.525 

5% 

5.375  000 

0.447  9 

92.868  9 

3% 

3.656  250 

0.304  7 

137.319 

513/32 

5.406  250 

0.450  5 

93.662  7 

3H16 

3.687  500 

0.307  3 

138.113 

57/16 

5.437  500 

0.453  1 

94.456  4 

323/32 

3.718  750 

0.309  9 

138.907 

515/32 

5.468  750 

0.455  7 

95.250  2 

3M 

3.750  000 

0.3125 

139.700 

5H 

5.500  000 

0.458  3 

LENGTHS. 


37 


Milli- 
meters. 

Inches. 

Feet. 

Milli- 
meters. 

Inches. 

Feet. 

140.494 

5i%2 

5.531  250 

0.460  9 

217.488 

89/16 

8.562  500 

0.7135 

141.288 

59/ie 

5.562  500 

0.463  5 

219.075 

8^ 

8.625  000 

0.7188 

142.082 

5i%2 

5.593  750 

0.466  1 

220.663 

8ii/i 

8.687  500 

0.724  0 

142.875 

5% 

5.625  000 

0.468  8 

222.250 

8M 

8.750  000 

0.729  2 

143.669 

5% 

5.656  250 

0.471  4 

223.838 

813/lfl 

8.812  500 

0.734  4 

144.463 

5ii/i6 

5.687  500 

0.474  0 

225.425 

87/8 

8.875  000 

0.739  6 

.145.257 

5*3/32 

5.718  750 

0.476  6 

227.013 

8i5/lfl 

8.937  500 

0.744  8 

146.050 

5H 

5.750  000 

0.479  2 

228.600 

9 

9.000  000 

0.750  0 

146.844 

525/32 

5.781  250 

0.481  8 

230.188 

9Vio 

9.062  500 

0.755  2 

147.638 

5*4i« 

5.812  500 

0.484  4 

231.775 

9^ 

9.125  000 

0.760  4 

148.432 

62%a 

5.843  750 

0.487  0 

233.363 

93/46 

9.187  500 

0.765  6 

149.225 

5y8 

5.875  000 

0.489  6 

234.950 

9M 

9.250  000 

0.770  8 

150.019 

52%2 

5.906  250 

0.492  2 

236.538 

95/16 

9.312  500 

0.776  0 

150.813 

515/10 

5.937  500 

0.494  8 

238.125 

m 

9.375  000 

0.781  3 

151.607 

5% 

5.968  750 

0.497  4 

239.713 

9Vl6 

9.437  500 

0.786  5 

152.400 

6 

6.000  000 

0.500  0 

241.300 

9H 

9.500  000 

0.791  7 

153.988 

6Vi6 

6.062  500 

0.505  2 

242.888 

99/16 

9.562  500 

0.796  9 

155.575 

6'/8 

6.125  000 

0.510  4 

244.475 

95/£ 

9.625  000 

0.802  1 

157.163 

63/16 

6.187  500 

0.5156 

246.063 

9H46 

9.687  500 

0.807  3 

158.750 

Q1A 

6.250  00( 

0.520  8 

247.650 

9M 

9.750  000 

0.812  5 

160.338 

65/16 

6.312  500 

0.526  0 

249.238 

913/16 

9.812  500 

0.817  7 

161.925 

Wi 

6.375  000 

0.531  3 

250.825 

V/s 

9.875  000 

0.822  9 

163.513 

6V16 

6.437  500 

0.536  5 

252.413 

915^6 

9.937  500 

0.828  1 

165.100 

VA 

6.500  000 

0.541  7 

254.001 

10 

10.000  000 

0.833  3 

166.688 

69/16 

6.562  500 

0.546  9 

255.588 

101/16 

10.062  500 

0.838  5 

168.275 

6^ 

6.625  000 

0.552  1 

257.176 

10H 

10.125  000 

0.843  8 

169.863 

611/16 

6.687  500 

0.557  3 

258.763 

103/16 

10.187  500 

0.849  0 

171.450 

6M 

6.750  000 

0.562  5 

260.351 

10& 

10.250  000 

0.854  2 

173.038 

6i%o 

6.812  500 

0.567  7 

261.938 

105/16 

10.312  500 

0.859  4 

174.625 

6^ 

6.875  000 

0.572  9 

263.526 

IO-H 

10.375  000 

0.864  6 

176.213 

6i5/16 

6.937  500 

0.578  1 

265.113 

lOJie 

10.437  500 

0.869  8 

177.800 

7 

7.000  000 

0.583  3 

266.701 

l<& 

10.500  000 

0.875  0 

179.388 

7Vio 

7.062  500 

0.588  5 

268.288 

109/16 

10.562  500 

0.880  2 

180.975 

7H 

7.125000 

0.593  8 

269.876 

10^ 

10.625  000 

0.885  4 

182.563 

73/10 

7.187  500 

0.599  0 

271.463 

ioiyi6 

10.687  500 

0.890  6 

184.150 

7M 

7.250  000 

0.604  2 

273.051 

1034 

10.750  000 

0.895  8 

185.738 

75/10 

7.312  500 

0.609  4 

274.638 

1013/16 

10.812  500 

0.901  0 

187.325 

7-Hi 

7.375  000 

0.614  6 

276.226 

10^ 

10.875  000 

0.906  3 

188.913 

7Vl6 

7.437  500 

0.619  8 

277.813 

1015/16 

10.937  500 

0.9115 

190.500 

7^ 

7.500  000 

0.625  0 

279.401 

11 

11.000000 

0.9167 

192.088 

79/16 

7.562  500 

0.630  2 

280.988 

HVl6 

11.062  500 

0.921  9 

193.675 

7% 

7.625  000 

0.635  4 

282.576 

iiH 

11.125000 

0.927  1 

105.203 

7H10 

7.687  500 

0.640  6 

284.163 

113/46 

11.187  500 

0.932  3 

106.850 

7''4 

7.750  000 

0.645  8 

285.751 

11M 

11.250000 

0.937  5 

108.438 

7'3/10 

7.812  500 

0.651  0 

287.338 

H5/16 

11.312500 

0.942  7 

200.025 

TX 

7.875  000 

0.656  3 

288.926 

11% 

11.375000 

0.947  9 

201.613 

715/io 

7.937  500 

0.661  5 

290.513 

llVie 

11.437500 

0.953  1 

203.200 

8 

8.000  000 

0.666  7 

292.101 

UH 

11.500000 

0.958  3 

204.788 

81/16 

8.062  500 

0.671  9 

293.688 

ii9/46 

11.562  500 

0.963  5 

206.375 

8H 

8.125  000 

0.677  1 

295.276 

UH 

11.625000 

0.968  8 

207.963 

Ss/ie 

8.187  500 

0.682  3 

296.863 

imie 

11.687500 

0.974  0 

209.550 

8^ 

8.250  000 

0.687  5 

298.451 

liH 

11.750000 

0.979  2 

211.138 

&%• 

8.312  500 

0.692  7 

300.038 

1113/16 

1.812500 

0.984  4 

212.725 

8-H 

8.375  000 

0.697  9 

301.626 

IV4 

1.875000 

0.989  6 

214.313 

8%« 

8.437  500 

0.703  1 

303.213 

lli5/46 

1.937500 

0.994  8 

215.900 

8^ 

8.500  000 

0.708  3 

304.801 

12 

2.000  000 

1.000  0 

38 


LENGTHS. 


Millimeters 

Feet  and  Inches. 

Millimeters. 

Feet  and  Inches. 

304.801 

1  foot  0  inches 

1  676.40 

5  feet  6  inches 

330.201 

1      1 

1  701.80 

5      7 

355.601 

1      2 

1  727.20 

5      8 

381.001 

1      3 

1  752.60 

5      9 

406.401 

1      4 

1  778.00 

5     10 

431.801 

1      5 

1  803.40 

5     11 

457.201 

1      6 

1  828.80 

6      0 

482.601 

1      7 

1  854.20 

6      1 

508.001 

1      8 

1  879.60 

6      2 

533.401 

1      9 

1  905.00 

6      3 

558.801 

1     10 

1  930.40 

6      4 

584.201 

1     11 

1  955.80 

6      5 

609.601 

2  feet  0 

1  981.20 

6      6 

635.001 

2      1 

2  006.60 

6      7 

660.401 

2      2 

2  032.00 

6      8 

685.801 

2      3 

2  057.40 

6      9 

711.201 

2      4 

2  082.80 

6     10 

736.601 

2      5 

2  108.20 

6     11 

762.002 

2      6 

2  133.60 

7     0 

787.402 

2      7 

2  159.00 

7      1 

812.802 

2      8 

2  184.40 

7     2 

838.202 

2      9 

2  209.80 

7      3 

863.602 

2     10 

2  235.20 

7      4 

889.002 

2     11 

2  260.60 

7      5 

914.402 

3      0 

2  286.00 

7     6 

939.802 

3      1 

2  311.40 

7     7 

965.202 

3      2 

2  336.80 

7     8 

990.602 

3      3 

2  362.20 

7     9 

1  016.00 

3      4 

2  387.60 

7     10 

1  041.40 

3      5 

2413.00 

7     11 

1  066.80 

3      6 

2  438.10 

8      0 

1  092.20 

3      7 

2  463.80 

8      1 

117.60 

3      8 

2  •^SO./'O 

8      2 

143.00 

3      9 

2  51  Uil 

8      3 

168.40 

3     10 

2  /il-P.Ol 

8      4 

193.80 

3     11 

2505.11 

8     5 

219.20 

4      0 

2  500.81 

8      6 

244.60 

4      1 

2  6  JO.  21 

8      7 

270.00 

4      2 

2  641.61 

8     8 

295.40 

4      3 

2  C.67.01 

8     9 

320.80 

4      4 

2  692.41 

8     10 

346.20 

4      5 

2  717.81 

8     11 

371.60 

4      6 

2  743.21 

9      0 

397.00 

4      7 

2  768.61 

9      1 

422.40 

4      8 

2  794.01 

9      2 

447.80 

4      9 

2819.41 

9      3 

473.20 

4     10 

2  844.81 

9      4 

498.60 

4     11 

2  870.21 

9      5 

1  524.00 

5      0 

2  895.61 

9      6 

549.40 

5      1 

2921.01 

9      7 

574.80 

5      2 

2  946.41 

9      8 

600.20 

5      3 

2971.81 

9      9 

625.60 

5      4 

2997.21 

9     10 

651.00 

5      5 

3  022.61 

9     11 

3  048.01 

10  feet  0 

LENGTHS. 
Conversion  Tables  for  Lengths. 


Ins.  = 

milim't 

Mnis  = 

inches. 

Feet  = 

meters. 

Mtrs  = 

feet..  .  . 

yards. 

Yds.= 

meters. 

Miles= 

klmtrs. 

Kims- 

miles. 

1 

25.400  1 

0.039  370 

0.304  801 

3.280  83 

0.914402 

1.09361 

1.60935 

.621  370 

2 

50.800  1 

0.078  740 

0.609  601 

6.561  67 

1.82880 

2.18722 

3.218  69 

1.24274 

3 

76.200  2 

0.118110 

0.914402 

9.84250 

2.74321 

3.28083 

4.828  04 

1.86411 

4 

101.600 

0.157480 

1.219  20 

13.1233 

3.657  61 

4.374  44 

3.437  39 

2.485  48 

5 

127.000 

0.196850 

1.52400 

16.4042 

4.57201 

5.468  06 

i.  046  73 

3.10685 

6 

152.400 

0.236  220 

1.82880 

19.6850 

5.48641 

6.561  67 

3.65608 

3.728  22 

7 

177.800 

0.275  590 

2.133  60 

22.965  8 

6.40081 

7.655  28 

11.2654 

4.34959 

8 

203.200 

0.314  960 

2.43840 

26.2467 

7.31521 

8.748  89 

12.8748 

4.97096 

9 

228.600 

0.354  330 

2.74321 

29.527  5 

8.229  62 

9.84250 

14.484  1 

5.59233 

10 

254.001 

0.393  700 

3.04801 

32.808  3 

9.14402 

10.936  1 

16.093  5 

6.21370 

11 

279.401 

0.433070 

3.35281 

36.089  2 

10.058  4 

12.029  7 

17.7028 

6.835  07 

12 

304.801 

0.472  440 

3.65761 

39.3700 

10.9728 

13.1233 

19.3122 

7.45644 

13 

330.201 

0.511810 

3.96241 

42.650  8 

11.8872 

14.2169 

20.921  5 

3.077  81 

14 

355.601 

0.551  180 

4.26721 

45.931  7 

12.801  6  . 

15.3106 

22.5309 

8.699  18 

15 

381.001 

0.590  550 

4.572  01 

49.2125 

13.7160 

16.4042 

24.1402 

9.320  55 

16 

406.401 

0.629  920 

4.87681 

52.493  3 

14.6304 

17.497  8 

25.749  6 

9.941  92 

17 

431.801 

0.669  290 

5.18161 

55.7742 

15.5448 

18.591  4 

27.3589 

10.5633 

18 

457.201 

0.708  660 

5.48641 

59.0550 

16.4592 

19.6850 

28.968  2 

11.1847 

19 

482.601 

0.748  030 

5.79121 

62.3358 

17.3736 

20.778  6 

30.577  6 

11.8060 

20 

508.001 

0.787  400 

6.09601 

65.6167 

18.2880 

21.8722 

32.1869 

12.4274 

21 

533.401 

0.826770 

6.40081 

68.897  5 

19.2024 

22.965  8 

33.796  3 

13.0488 

22 

558.801 

0.866  140 

6.70561 

72.1783 

20.1168 

24.059  4 

35.405  6 

13.670  1 

23 

584.201 

0.905510 

7.01041 

75.459  2 

21.0312 

25.153  1 

37.015  0 

14.2915 

24 

303.601 

0.944880 

7.31521 

78.7400 

21.9456 

26.2467 

38.624  3 

14.9129 

25 

635.001 

0.984  250 

7.62002 

82.020  8 

22.860  0 

27.3403 

40.233  7 

15.534  2 

26 

660.401 

1.02362 

7.92482 

85.3017 

23.774  4 

28.4339 

41.8430 

16.1556 

27 

685.801 

.062  99 

8.229  62 

88.582  5 

24.688  9 

29.527  5 

13.452  4 

16.7770 

28 

711.201 

.10236 

8.534  42 

91.8633 

25.603  3 

30.621  1 

15.061  7 

17.398  4 

29 

736.601 

.14173 

8.83922 

95.1442 

26.517  7 

31.7147 

16.671  1 

18.0197 

30 

762.002 

.181  10 

9.14402' 

98.425  0 

27.432  1 

32.808  3 

48.280  4 

18.641  1 

31 

787.402  ' 

.220  47 

9.44882 

101.706 

28.3465 

33.901  9 

49.889  8 

19.2625 

32 

812.802 

.259  84 

9.753  62 

104.987 

29.2609 

34.995  6 

51.499  1 

19.8838 

33 

838.202 

.299  21 

10.058  4 

108.268 

30.1753 

36.089  2 

53.1085 

20.505  2 

34 

863.602 

1.33858 

10.3632 

111.548 

31.0897 

37.182  8 

54.7178 

21.1266 

35 

889.002 

1.37795 

10.6680 

114.829 

32.004  1 

38.276  4 

56.327  2 

21.7479 

36 

914.402 

1.41732 

10.9728 

118.110 

32.9185 

39.3700 

57.9365 

22.369  3 

37 

139.802 

1.45669 

11.2776 

121.391 

33.8329 

40.463  6 

59.545  8 

22.990  7 

38 

365.202 

1.43606 

11.5824 

124.672 

34.747  3 

41.5572 

51.1552 

23.612  1 

39 

990.602 

1.53543 

11.8872 

127.953 

35.661  7 

42.6508 

62.7645 

24.233  4 

40 

1  016.00 

1.57480 

12.1920 

131.233 

36.576  1 

43.744  4 

64.3739 

24.8548 

41 

1  041.40 

1.61417 

12.4968 

134.514 

37.4905 

44.838  1 

65.983  2 

25.4762 

42 

1  066.80 

1.65354 

12.8016 

137.795 

38.4049 

45.931  7 

67.592  6 

26.097  5 

43 

1  092.20 

1.69291 

13.1064 

141.076 

39.3193 

47.025  3 

69.2019 

26.7189 

44 

1  117.60 

1.73228 

13.4112 

144.357 

40.233  7 

48.1189 

70.8113 

27.3403 

45 

1  143.00 

1.77165 

13.7160 

147.638 

41.1481 

49.2125 

72.420  6 

27.961  6 

46 

1  168.40 

1.81102 

14.0208 

150.918 

42.062  5 

50.306  1 

74.0300 

28.583  0 

47 

1  193.80 

1.85039 

14.3256 

154.199 

42.9769 

51.3997 

75.639  3 

29.204  4 

48 

1219.20 

1.88976 

14.6304 

157.480 

43.891  3 

52.4933 

77.248  7 

29.8258 

49 

1  244.60 

1.92913 

14.9352 

160.761 

44.805  7 

53.5869 

78.858  0 

30.447  1 

50 

1  270.00 

1.96850 

15.2400 

164.042 

45.720  1 

54.680  6 

80.467  4 

31.0685 

40  LENGTHS. 

Conversion  Tables  for  Lengths   (CONCLUDED). 


Ins.  = 

milimM 

Mms  = 

inches. 

Feet  = 

meters. 

Mtrs  = 

feet  .  . 

yards. 

Yds.= 

meters. 

Miles= 

klmtrs. 

Klms= 

miles. 

51 

1  295.40 

2.007  87 

15.5448 

167.323 

46.634  5 

55.774  2 

82.0767 

31.6899 

52 

1  320.80 

2.047  24 

15.849  6 

170.603 

47.5489 

56.807  8 

83.686  1 

32.3112 

53 

1  346.20 

2.08661 

16.1544 

173.884 

48.463  3 

57.9014 

85.295  4 

32.932  6 

54 

1371.60 

2.12598 

16.4592 

177.165 

49.3777 

59.055  0 

80.904  7 

33.5540 

55 

1  397.00 

2.16535 

16.7640 

180.446 

50.292  1 

00.148  6 

88.514  1 

34.1753 

56 

1  422.40 

2.20472 

17.0688 

183.727 

51.2065 

61.2422 

90.1234 

34.796  7 

57 

1  447.80 

2.24409 

17.3736 

187.008 

52.1209 

62.3358 

91.7328 

35.418  1 

58 

1  473.20 

2.283  46 

17.6784 

190.288 

53.035  3 

63.429  4 

93.342  1 

36.039  5 

59 

1  498.60 

2.32283 

17.9832 

193.569 

53.949  7 

64.523  1 

94.951  5 

36.660  8 

60 

1  524.00 

2.362  20 

18.2880 

196.850 

54.864  1 

65.6167 

90.5008 

37.282  2 

61 

1  549.40 

2.40157 

18.5928 

200.131 

55.7785 

66.7103 

98.1702 

37.903  6 

62 

1  574.80 

2.44094 

18.897  6 

203.412 

56.692  9 

07.8039 

99.779  5 

38.524  9 

63 

1  600.20 

2.48031 

19.2024 

206.693 

57.607  3 

68.897  5 

101.389 

39.1463 

64 

1  625.60 

2.51968 

19.5072 

209.973 

58.5217 

09.991  1 

102.998 

39.767  7 

65 

1  651.00 

2.559  05 

19.8120 

213.254 

59.436  1 

71.0847 

104.608 

40.389  0 

66 

1  676.40 

2.598  42 

20.1168 

216.535 

60.350  5 

72.1783 

106.217 

41.0104 

67 

1701.80 

2.63779 

20.421  6 

219.816 

61.2649 

73.2719 

107.826 

41.6318 

68 

1  727.20 

2.677  16 

20.7264 

223.097 

62.1793 

74.3050 

109.436 

42.253  2 

69 

1  752.60 

2.71653 

21.0312 

226.378 

63.093  7 

75.459  2 

111.045 

42.8745 

70 

1  778.00 

2.75590 

21.3360 

229.658 

64.008  1 

76.552  8 

112.654 

43.495  9 

71 

1  803.40 

2.795  27 

21.6408 

232.939 

64.922  5 

77.6464 

114.264 

44.1173 

72 

1  828.80 

2.834  64 

21.9456 

236.220 

65.8369 

78.7400 

115.873 

44.738  6 

73 

1  854.20 

2.87401 

22.2504 

239.501 

66.751  3 

79.833  6 

117.482 

45.3600 

74 

1  879.60 

2.91338 

22.555  2 

242.782 

67.665  7 

80.927  2 

119.092 

45.981  4 

75 

1  905.00 

2.952  75 

22.8600 

246.063 

68.580  1 

82.020  8 

120.701 

46.602  7 

76 

1  930.40 

2.992  12 

23.1648 

249.343 

69.4945 

83.1144 

122.310 

47.224  1 

77 

1  955.80 

3.031  49 

23.469  6 

252.624 

70.408  9 

84.208  1 

123.920 

47.8455 

78 

1981.20 

3.07086 

23.7744 

255.905 

71.3233 

85.301  7 

125.529 

48.466  9 

79 

2  006.60 

3.11023 

24.079  2 

259.186 

72.2377 

80.395  3 

127.138 

49.0882 

80 

2  032.00 

3.14960 

24.3840 

262.467 

73.1521 

87.488  9 

128.748 

49.709  6 

81 

2  057.40 

3.18897 

24.688  8 

265.748 

74.0065 

88.5825 

130.357 

50.331  0 

82 

2  082.80 

3.228  34 

24.9936 

269.028 

74.981  0 

89.070  1 

131.966 

50.952  3 

83 

2  108.20 

3.26771 

25.298  4 

272.309 

75.8954 

90.709  7 

133.576 

51.5737 

84 

2  133.60 

3.30708 

25.603  3 

275.590 

76.809  8 

91.8033 

135.185 

52.1951 

85 

2  159.00 

3.34645 

25.908  1 

278.871 

77.7242 

92.9509 

136.795 

52.8164 

86 

2  184.40 

3.38582 

26.2129 

282.152 

78.638  6 

94.050  0 

138.404 

53.4378 

87 

2  209.80 

3.425  19 

26.5177 

285.433 

79.553  0 

95.1442 

140.013 

54.0592 

88 

2  235.20 

3.46456 

26.8225 

288.713 

30.467  4 

90.2378 

141.623 

54.680  6 

89 

2260.60 

3.50393 

27.1273 

291.994 

81.3818 

97.331  4 

143.232 

55.301  9 

90 

2  286.00 

3.54330 

27.432  1 

295.275 

82.2962 

98.425  0 

144.841 

55.923  3 

91 

2311.40 

3.58267 

27.7369 

298.556 

83.2106 

99.5180 

140.451 

56.544  7 

92 

2  336.80 

3.62204 

28.041  7 

301.837 

84.1250 

100.012 

148.000 

57.1660 

93 

2  362.20 

3.66141 

28.3465 

305.118 

85.039  4 

101.700 

149.009 

57.787  4 

94 

*>  387.60 

3.700  78 

28.651  3 

308.398 

85.953  8 

102.799 

151.279 

58.408  8 

95 

2413.00 

3.740  15 

•>8.956  1 

311.679 

86.868  2 

103.893 

152.888 

59.030  1 

96 

2438.40 

3.77952 

'9.2609 

314.960 

87.782  6 

104.987 

154.497 

59.651  5 

97 

2  463.80 

3.81889 

9.565  7 

318.241 

88.6970 

100.080 

150.107 

60.272  9 

98 

489.20 

3.858  26 

9.8705 

321.522 

89.6114 

107.174 

157.716 

60.894  3 

99 

514.60 

3.897  63 

0.1753 

324.803 

90.525  8 

108.208 

159.325 

61.5150 

100 

2  540.01 

3.937  00 

0.480  1 

328.083 

91.4402 

109.301 

160.935 

62.1370 

SURFACES.  41 


SURFACES. 

There  are  no  fundamental  standards  of  surfaces  or  areas,  as  these  meas- 
ures are  all  based  on  the  linear  measures,  which  see  for  their  fundamental 
values.  The  legal  U.  S.  yard,  foot,  inch,  mile,  etc.,  being  about  3  parts 
in  one  million  larger  than  the  legal  values  in  Great  Britain,  the  U.  S.  square 
yard,  square  foot,  etc.,  will  be  about  6  parts  in  one  million  larger  than 
the  British,  a  difference  which  is  absolutely  negligible  in  all  but  the  most 
refined  physical  measurements.  The  values  given  in  the  following  tables 
are  based  on  the  U.  S.  legal  relation.  To  reduce  the  values  in  these  tables 
to  British  square  miles,  square  yards,  square  feet,,and  similar  units,  when 
very  great  accuracy  is  required  subtract  6  parts  for  every  million  from 
all  those  values  of  a  sq.  mile,  sq.  yard,  etc.,  which  are  in  terms  of  other 
units  than  miles,  yards,  etc.;  for  instance,  add  this  correction  in  the  value 
of  one  sq.  mile  in  sq.  meters,  but  not  in  sq.  feet,  as  the  latter  is  the  same 
for  both ;  or  add  6  parts  for  every  million  to  all  the  values  of  other  units 
which  are  given  in  terms  of  sq.  miles,  sq.  yards,  etc.  This  will  never  affect 
the  5th  place  of  figures  by  more  than  one  unit  and  generally  less. 

A  circular  unit  or  measure  (such  as  a  circular  mil)  is  the  area  of  a 
circle  whose  diameter  is  one  unit  or  measure  (as  one  mil).  It  is  used  for 
cross-sections  of  round  wires,  pipes,  rods,  etc.,  and  avoids  the  necessity 
of  using  the  value  of  ir  (  =  3.141  592  65).  If  the  diameter  of  a  circle  is  d, 
its  area  in  circular  units  is  simply  d2.  Some  of  the  specific  relations  given 
in  the  table  may  also  be  expressed  as  follows: 

If  d  is  the  diameter  of  a  circle  expressed  in  one  kind  of  a  unit,  then  the 
area  in  the  other  unit  will  be: 

if  d   s  in  mils  the  area  in  sq.  millimeters      =d2X  0.000  506  709* 


if  d 
itd 
if  d 
if  d 
itd 
itd 
itd 


s  in  millimeters  the  area  in  sq.  mils      =d2X  1  217.36* 
s  in  centimeters  the  area  in  sq.  inches  =  d2X 0.121  736* 
s  in  centimeters  the  area  in  sq.  feet      =  d2  X  0.000  845  391* 
s  in  inches  the  area  in  sq.  centimeters  =  d2X  5.067  09* 
s  in  inches  the  area  in  sq.  feet  =d2X 0.005  454  15* 

s  in  feet  the  area  in  sq.  centimeters      =  d2X729.662* 
s  in  feet  the  area  in  sq.  inches  =  d2X  113.097* 


SURFACES;     Circular  or   Cross -section   Measures.     Usual. 

Aprx.  means  within  2%  ;  "sq."  means  square  and  is  often  used  as  a  suffix 
instead  of  the  exponent  (2),  being  simpler  for  printing  and  type-writing; 
thus  sq.  cm  for  era2;  sq.  ft  for  ft2. 

*  Checked  by  L.  A.  Fischer,  Asst.  Phys.  National  Bureau  of  Standards. 

1  circular  mil  [CM]=         0.785  398*  sq.  mil.    Aprx.  %0.  -  .  • 

-0.000  645  163*  Cmm.    Aprx.  ifa  -*- 1  000  .. 
=  0.000  506  709*  sq.  mm.    Aprx.  V£-f- 1  000 . 

=         0.000  001    circular  inch 

1  sq.  mil  [mil2]  =  1.273  24*  circular  mils.    Aprx.  1% 

=  0.000  821  447*  circ.  mm.    Aprx.  Vi2  •*•  100 

=  0.000645  163*  sq.  millimeter.  Aprx.  %i-*-l  000  J_. 

=         0.000  001    sq.  inch 6-000  0000 

1  circular  millimeter  [Cmm]: 

=    1  550.00*  circular  mils.    Aprx.  i#  X  1  000 3.190  3308 

=    1  217.36*  sq.  mils.    Aprx.  %  X  1  000 3.085  4207 

=  0.785  398*  sq.  millimeter.    Aprx.  %0 1-895  0899 

=           0.01    circular  centimeter 2-000  0000 

1  sq.  millimeter  [mm2]  =         1  973.52*  cir.  mils.    Aprx.  2  000. .  3.295  2409 

=         1  550.00*  sq.  mils.  Ap.  1%  X  1  000  3-190  3308 

=         1.273  24*  circ.  mm.    Aprx.  1%.  .  .  0-104  9101 

=   0.012  732  4*  circ-.  cm.    Aprx.  Ho 2-104  9101 

=                0.01    sq.  centimeter 2-000  0000 

=0.001  550  00*  sq.  in.    Aprx.  %8  •*•  100  .  5.190  3308 


42  SURFACES. 

1  circular  centimeter  [Ccm]: 

=  155  000.*  circular  mils.    Aprx.  Ufa  X  100  000 5-190  3308 

121  736.*  sq.  mils.    Aprx.  12  X  10  000 : 5-085  4207 

=  100.    circular  millimeters 2-000  0000 

78.539  8  *  sq.  millimeters.    Aprx.  80 1-895  0899 

=         0.785  398  *  sq.  centimeter.    Aprx.  8/lo 1-895  0899 

=         0.155  000  *  circular  inch.    Aprx.  %3 1-190  3308 

=         0.121  736  *  sq.  inch.    Aprx.  12-^100 1-085  4207 

=   0.001  076  39  *  circular  foot.    Aprx.  108-^-100  000 3-031  9683 

=  0.000  845  391  *  sq.  foot.    Aprx.  1/12-^  100 4-927  0582 

1  sq.  centimeter  [cm2]: 

=  197  352.*  circular  mils.    Aprx.  2  X  100  000 5-295  2409 

155  000.*  sq.  mils.    Aprx.  *Vr  X  100  000 5-190  3308 

127.324  *  circular  millimeters.    Aprx.  l/$X  1000 2-1049101 

100.    sq.  millimeters 2. 000  0000 

1.273  21  *  circular  centimeters.    Aprx.  1% 0-104  9101 

=         0.197  352  *  circular  inch.    Aprx.  ^0- .  - 1-295  2409 

=         0.155  000  *  sq.  inch.    Aprx.  %8 1-190  3308 

=  0.01  *  sq.  decimeter 2-000  0000 

=   0.001  370  50  *  circular  foot.    Aprx.  1!  *-s- 1  000 3-136  8784 

=  0.001  076  387  *  sq.  foot.    Aprx.  108 -=-100  000 3-031  9683 

t  circular  inch  [Gin]: 

=       1  000  000.    circular  mils 6-000  0000 

=          785  398.*  sq.  mils.    Aprx.  %  X  1  000  000 5-895  0899 

=          645.103  *  circular  millimeters.    Aprx.  1%  X  100 2-809  6692 

=          506.709  *  sq.  millimeters.    Aprx.  ^X  1  000 2-704  7591 

=         6.451  63  *  circular  centimeters.    Aprx.  *% 0-809  6692 

=         5.067  09  *  sq.  centimeters.    Aprx.  5 0-704  7591 

=       0.785  398  *  sq.  inch.    Aprx.  %0 1-895  0899 

=  0.006  944  44  *  circular  foot.    Aprx.  7-*- 1  000 3-841  6375 

=  0.005  454  15  *  sq.foot.    Aprx.  «/n-v- 100 3-736  7274 

I  sq.  inch  [in2]=       1  273  240.*  cir.  mils.    Aprx.  HX  10000  000  6-104  9101 

=       1  000  000.    sq.  mils 6-000  0000 

=          821.447  *  cir.  mm.    Aprx.  %X  1  000 2-914  5793 

=          645.163  *  sq.  mm.    Aprx.  1%  X  100 2-809  6692 

=         8.214  47  *  cir.  centimeters.    Aprx.  %  X  10. .   0-914  5793 

=         6.451  63  *  sq.  cm.    Aprx.  1%  or  6M 0-809  6692 

=         1.273  24  *  circular  inches.    Aprx.  1% 0-104  9101 

=   0.064  516  3  *  sq.  decimeter.    Aprx.  %i-i-  10....   2-809669? 
=  0.008841  94  *  circular  foot.    Aprx.  9-=- 1  000..  .   3-9465476 

=  0.006  944  44  *  sq.  foot.    Aprx.  ftooo 3-841  6375 

I  sq.  decimeter  [dm2]  =  100.  sq.  centimeters 2-000  0000 

=       15.5000*  sq.  inches.  Aprx.  iVrX  10.    1.1903308 
=  0.107  638  7*  sq.foot.    Aprx.  11  -=-  100..   1-031  968? 

=  0.01    sq.  meter 2-000  000( 

=  0.011  9599*  sq.  yard.  Aprx.  12-4-1000    2-0777253 
1  circular  foot  [Cft]  =     929.034*  cir.  cm.    Aprx.  1^2  X  1  000..  .   2-968  0317 

=     729.662*  sq.  cm.    Aprx.  s/u  X  1  000 2-863  1216 

144.  circular  ins.  Aprx.  M- X  1000.   2-1583625 
=     113.097*  sq.  inches.    Aprx.  %  X  1  000.  .    2-0534524 

=  0.785  398*  sq.  foot.    Aprx.  %0 1-895  0893 

1  sq.  foot  [ft2]  (Brit.)  =  0.999  994  2  sq.  foot  (U.  S.).    Aprx.  1 L999  9975 

=  0.092902  9  sq.  meter.    Aprx.  Hia-*- 10.  .   2-9680292 

1  sq.foot  [ft2]  (U.S.)  =       1  182.88*  cir.  cm.    Aprx.  %X  1  000.  .    3-0729410 
=        929.034*  sq.  cm.    Aprx.  n/12  X  1  000.  2-968  0317 

=        183.346*  cir.  in.    Aprx.  ^X  100 2-2632723 

=  144.  sq.  ins.    Aprx.  V7  X  1  000.  ..   2-158  3625 

=  9.290  34*  sq.  dm.  Aprx.  H12X  10.  ..  0-968  0317 
=  1.273  24*  circular  feet.  Aprx.  1%.  .  ..  0-104  9101 
=  1.000005  7  sq.  feet  (Brit.).  Aprx.  1..  .  0-000  0025 

=     0.111  111*  sq.  yard,  or  V9 1-0457575 

=  0.092  903  4*  sq.  meter.  Aprx.  iMa-s-10.   2-968  0317 

Isq.  yard  [yd2]  (Brit. )=  0.999  994  3  sq.  yard  (U.S.).  Aprx.  1...  1-999  9975 
=    0.836  126  sq.  meter.    Aprx.% 1.922  2717 


SURFACES.  43 


1  sq.  yard[yd2](U.  S.)  =           8  361.31  *  sq.  cm.  Ap.%X  10000  3-922  2742 

1  296.*  sq.  ins.  Aprx.  13 X  100  3. 112  6050 

=           83.613  1  *  sq.  dm.  Aprx.  %X100  1-922  2742 

9.    sq.feet 0-9542425 

=      1.0000057      sq.  yds.   (Brit.).  Ap.  1.  0-000  0025 

=         0.836  131  *  sq.  meter.     Aprx.  %  .  .  f.922  2742 

=   0.008  361  31  *  are.  Aprx.  %-s-100 3.922  2742 

=  0.000  206  612  *  acre.  Ap.  21  -*- 100  000.  4.315  1546 

1  sq.  meter  [m2]  =               10  000.    sq.  centimeters 4-000  0000 

=           1  550.00  *  sq.  ins.    Aprx.  i#  X  1  000. ...  3-1903308 

=                    100.    sq.  decimeters 2-000  0000 

=         10.76393     sq.ft.  (Brit.).  Aprx.  12Ai  X  10  1-031  9708 

=         10.763  87  *  sq.  ft.  (U.  S.).  Aprx.  i%i  X  10  1.Q31  9683 

=           1.19599     sq.  yds.  (Brit.).  Aprx.  add  %  0-0777283 

=           1.195  99  *  sq.  yds.  (U.S.).  Aprx.  add  %  0-0777258 

=                   0.01      are 2-000  0000 

=  0.000  247  104  *  acre.    Aprx.  \i  +  \  000 4-392  8804 

1  are  or  ar  [a]  =     1  076.387*  sq.  feet.    Aprx.  i%i  X  1  000 3-031  9683 

=        119.599*  sq.  yards.    Aprx.  120 2-077  7258 

=                100.  sq.  meters 2-000  0000 

=                  10.  meters  square 1-000  0000 

"              =                    1 .  sq.  decameter 0-000  0000 

=  0.024  710  4*  acre.    Aprx.M^-10 2-3928804 

=              0.01    hectare 2-000  0000 

1  acre  [A]  =            43  560.*  sq.  feet.    Aprx.  HsX  1  000  000 4-639  0879 

=              4  840.*  sq.  yards.    Aprx.  4  800 3-684  8454 

=         4  046.87  *  sq.  meters.    Aprx.4XlOOO 3-6071196 

=          208.710  *  feet  square.    Aprx/210 2-319  5440 

=         40.468  7  *  ares.    Aprx.  40 1-6071196 

=       0.404  687  *  hectare.    Aprx.  y10 1-607  1196 

=  0.00404687  *  sq.  kilometer.    Aprx.  4 -f-  1000 3-6071196 

==0.001  562  50*  sq.  mile.    Aprx.  i#-s- 1  000 3-1938200 

1  hectare  [ha]=       107  638.7*  sq.  feet.    Aprx.  i2/lt  x  100  000.  .  ..  5-031  9683 

=         11  959.9*  sq.  yards.    Aprx.  12  X  1  000 4-077  7258 

=             10  000.  sq.  meters 4-000  0000 

=                  100.  ares 2-000  0000 

=         2.471  04*  acres.    Aprx.  1% 0-392  8804 

=                 0.01    sq.  kilometer 2-000  0000 

=  0.003861  01*  sq.  mile.    Aprx.  ^6-^10 3-586  7004 

lsq.kilometer[km2]=  10763  867.*  sq.ft.  Ap.  i%±  X  10  000000  7-031  9683 
=  1  195  985.*  sq.  yds.  Aprx.  12  X  100  000  6-077  7258 

=  1  000  000.    sq.  meters 6-000  0000 

=        10  000.    ares 4-000  0000 

=      247.104  *  acres.    Aprx.  MX  1  000..  ..  2-392  8804 

=             100.    hectares 2-000  OOOC 

=  0.386  101  *  sq.  mile.    Aprx.  Ke  X  10.  .  .  1-586  7004 


1  sq.  mile  [ml2]  =  27  878  400. 

=  3097600. 

=  2589999. 

=  25  900.0 

=  640. 

=  259.000 

=  2.590  00 


sq.  feet.    Aprx.  ^  X  10  000  000  .  7-445  2678 

sq.  yards.    Aprx.  31  X  100  000.  .  .  6-491  0253 

sq.  meters.    Aprx.  26  X  100  000 .  .  6-413  2996 

ares.    Aprx.  26  X  1  000 4-413  2996 

acres 2-806  1800 

hectares.    Aprx.  260 2-413  2996 

sq.  kilometers.    Aprx.  26  -^  10 0-413  2996 


SURFACES  (continued).     Unusual,  Special  Trade,  or 
Obsolete. 

1  milliare  =  0.1  sq.  meter. 

1  ceii tare,  centar,  or  centaire  =  10.763  87  sq.  feet  =  l.  sq.  meter. 

1  square  (building)  =  100.  sq.  feet. 

1  declare  =  10.  sq.  meters  =  0.1  are. 

1  sq.  rod,  or  sq.  pole,  or  sq.  perch  =  625.  sq.  links  (survey or 's)  =  272M 
sq.  feet  =  30M  sq-  yards  =  25. 293  sq.  meters  =  Vi60  acre. 

1  sq,  chain  (Gunter's  or  surveyor's)  =  4  356.  sq.  feet  =  484.  sq.  yards  = 
404.687  sq.  meters  =  16.  sq.  rods,  poles,  or  perches  =  4.046  87  ares  =  Mo  acre  = 
0.0404687  hectare  =  0.000  404  687  sq.  kilometer  =  0.000  156  250  sq.  mile. 


44 


SURFACES. 


1  sq.  meter  =  0.002  471  04  sq.  chain.  1  are  =  0.247  104  sq.  chain.  1  acre  = 
10.  sq.  chains.  1  hectare  =  24. 7 10  4  sq.  chains. 

1  decare  (not  used)  =  l  000.  sq.  meters  =10.  ares. 

1  rood  ("R]=10  890.  sq.  feet  =  1  210.  sq.  yards  =  1  011.72  sq.  meters  = 
40.  sq.  rods,  poles,  or  perches  =  3^  acre. 

l.fardingdeal  (Brit.)-l.  rood. 

1  circular  acre  =  235.504  feet  diameter. 

1  section  (of  land)  =  1.  mile  square. 

1  township  =  36.  sq.  miles. 

1  sq.  myriameter  or  miriameter  =  100.  sq.  kilometers  =  38.610  1  sq. 
miles. 

SURFACES  (concluded).     Foreign. 

These  are  mostly  obsolete  as  the  metric  system  is  now  used  in  most 
foreign  countries.  The  British  measures  are  included  among  the  U.  S. 
measures,  being  very  nearly,  and  sometimes  quite,  the  same.  The  trans- 
lated terms  are  merely  synonymous,  and  not  the  exact  equivalents. 

Germany.  Prussia  1  Morgen=180  sq.  Ruthen  (rods).  1  Morgen  = 
25.53225  ares  =  2  553.225  sq.  meters  =  0.630  912  acres.  According  to 
Nystrom's  Mechanics,  1  Berlin  Morgen,  great,  =  6  786  sq.  yards;  ditto, 
small,  =  3  054  sq.  yards.  1  Hamburg  Morgen  =11  545  sq.  yards.  1  Han- 
over Morgen  =  3  100  sq.  yards.  1  Prussian  Morgen  =  3  053  sq.  vards. 

France.  1  arpent  (acre)  =  100  sq.  perches  (French,  nearly  same  as 
U.  S.)  =  34.19  are,  or  51.07  are  =  0.844  849  acres,  or  1.261  96  acres. 

Austria.  1  Joch=1600  sq.  Klafter  =  5755  sq.  meters  =  1.422  08 
acres.  1  Vienna  Joch  =  6  889  sq.  yards. 

Sweden.  1  Tunnland  =  56  000  sq.  fot  =  49.364  1  ares=1.21981  acres. 
According  to  Nystrom  it  is  equal  to  5  900  sq.  yards. 

Russia.  1  Dessaetine  =  2  400  sq.  saschehn  =  10  925  sq.  meters= 
2.699  61  acres.  13  066.2  sq.  yards  (Nystrom). 

Switzerland.  1  Geneva  arpent  =  6  179  sq.  yards;  1  faux  =  7  855  sq. 
yards;  1  Zurich  acre  =  3  875.0  sq.  yards. 

Spain.  1  fanegada  (since  1801)  =  !. 5871  acres  =  69  134.08  sq.  feet; 
according  to  Nystrom  it  is  equal  to  5  500  sq.  yards. 

Japan.  1  cho  (apparently  not  1  cho  length  squared)  =  10  tan  of 
10  se  of  30  tsubo.  1  tsubo  =  l  ken  square  =  3. 31  square  meters  =  35. 6 
square  feet.  1  square  meter  =  0.302  tsubo.  1  square  foot  =  0.028  1  tsubo. 
1  cho  =  2. 45  acres. 

Miscellaneous.  (From  Nystrom's  Mechanics.)  1  fanegada,  Canary 
Isles,  =  2422  sq.  yards.  1  acre,  Ireland,  =  7  840  sq.  yards.  1  acre,  Scot- 
land, =6  150  sq.  yards.  1  moggia,  Naples,  =  3998  sq.  yards.  1  Pezza, 
Rome,  =  3  158  sq.  yards.  1  Geira,  Portugal,  =  6  970  sq.  yards. 

Conversion  Tables  for  Surfaces. 


Sq.  in.= 
Sq.cm.= 
Sq.  ft.= 
Sq.  m.= 
Sq.yd.= 
Acres  =* 
Hect'r.- 

sq.  cm. 

sq.  ins. 

sq.  met. 

sq  ft 

sq.  yds. 

hect'res. 

acres. 

sq.met. 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

6.451  63 
12.903  3 
19.3549 
25.806  5 
32.258  1 
38.709  8 
45.1614 
51.6130 
58.064  6 
64.5163 

0.155000 
0.309  999 
0.464999 
0.619999 
0.774  998 
0.929  998 
1.08500 
1.23999 
1.39500 
1.55000 

0.092  903 
0.185807 
0.278  710 
0.371  614 
0.464517 
0.557  420 
0.650  324 
0.743227 
0.836  131 
0.929  034 

10.763  9 
21.5277 
32.291  6 
43.0555 
53.8193 
64.583  2 
75.347  1 
86.1110 
96.874  8 
107.639 

0.836  13 
1  67226 
2.508  39 
3.344  52 
4.18065 
5.01678 
5.85291 
6.689  05 
7.525  18 
8.36131 

1.19599 
2.391  97 
3.58796 
4.783  94 
5.979  93 
7.17591 
8.371  90 
9.567  88 
10.7639 
11.9599 

0.404  687 
0.809  375 
1.21406 
1.61875 
2.023  44 
2.428  12 
2.83281 
3.237  50 
3.642  19 
4.04687 

2.471  04 
4.942  09 
7.41313 
9.884  18 
12.3552 
14.8263 
17.297  3 
19.768  4 
22.239  4 
24.7104 

VOLUMES.  45 


VOLUMES.     Cubic  and  Capacity  Measures. 

The  fundamental  standard  measures  of  volumes  in  the  United  States 
are;  (a)  the  cubes  of  linear  dimensions  in  terms  of  units  based  on  the 
international  meter;  (b)  the  liter  which  is  the  volume  of  the  mass  of  one 
international  kilogram  of  pure  water  at  its  maximum  density,  and  at 
760  mm.  barometric  pressure;  (c)  the  gallon,  which  is  equal  to  exactly 
231  cubic  inches;  (d)  the  bushel  (the  old  Winchester  bushel  of  England) 
which  is  equal  to  exactly  2  150.42  cubic  inches.  The  inch  here  referred 
to  is  the  one  derived  from  this  meter.  The  liter,  being  based  on  this  kilo- 
gram, is  the  same  in  all  the  countries  participating  in  the  international  con- 
vention. These  four  measures  of  volumes  and  capacities  are  those  now  used 
by  the  National  Bureau  of  Standards  in  Washington  and  are  in  general  use 
in  this  country.  They  are  definite  and  accurate,  except  that  the  pre- 
cise volume  of  the  liter  expressed  in  cubic  centimeter  is  still  slightly  un- 
certain as  explained  below. 

The  liter  is,  according  to  the  decision  some  years  ago  of  the  Interna- 
tional Committee  of  Weights  and  Measures,  defined  as  "the  volume  of  the 
mass  of  one  kilogram  of  pure  water  at  its  maximum  density,  and  under 
normal  atmospheric  pressure."  The  accepted  temperature,  at  present, 
at  which  the  density  of  water  is  a  maximum,  is  4°  C.,  but  this  may  be 
altered  slightly  by  subsequent  determinations.  The  normal  atmospheric 
pressure  referred  to  is  that  exerted  by  a  vertical  column  of  mercury 
760  mm.  high,  at  latitude  45°  and  at  sea  level,  the  mercury  having  a  den- 
sity of  13.595  93.  This  definition  of  the  liter  has  been  adopted  by  the 
National  Bureau  of  Standards.  It  was  originally  intended  that  the  kilo- 
gram should  be  the  weight  of  a  cubic  decimeter  of  water,  thus  making 
the  liter  exactly  a  cubic  decimeter.  The  kilogram,  however,  being  now 
fixed  definitely  as  the  weight  of  a  certain  piece  of  metal,  and  as  this  estab- 
lishes the  liter,  it  still  remains  to  determine  by  measurement  the  precise 
volume  of  the  liter  in  cubic  decimeters.  The  International  Bureau  began 
this  determination  some  five  years  ago,  and  the  results  thus  far  obtained 
indicate  that  the  liter  is  about  25  parts  in  1  000000.  greater  than  the  cubic 
decimeter. f  As  this  relation  will  always  be  subject  to  slight  corrections, 
jvnd  as  the  difference  is  of  no  consequence  in  practice,  the  National  Bureau 
of  Standards  has  for  the  present  assumed  that  the  liter  and  the  cubic 
decimeter  are  equivalent;  this  identity  is  assumed  also  in  all  the  tables 
in  this  book.  The  difference  above  mentioned  would  at  most  affect  only 
the  fifth  place  of  figures  slightly*  and  generally  only  the  sixth.  The  capac- 
ities of  vessels  are  determined  by  weighing  the  water  necessary  to  fill 
them,  and  not  by  measuring  their  dimensions  (see  the  table  of  the  weights 
and  volumes  of  water). 

The  Customs  Service  and  the  Internal  Revenue  Bureau  uss  the  gallon 
of  231  cubic  inches  above  referred  to;  it  is  the  old  British  wine  gallon. 
In  these  tables  this  gallon  and  the  measures  based  on  it  are  indicated  as 
"liquid;  U.  S."  to  distinguish  them  from  the  "dry;  U.  S."  measures  and 
the  Imperial  or  "Brit.'*  measures. 

The  gallon  and  liter  are  the  units  for  measuring  liquids,  while  the  bushel 
and  liter  are  the  units  for  dry  measures,  as  for  grain,  fruits,  vegetables, 
etc.  The  "dry"  measures  are  about  one  sixth  greater  than  the  corre- 
sponding "liquid"  measures;  accurately: 

dry  measures XI.  163  65  (aprx.  add  ^)  =  liquid  meas.  (log  0-065  8213) 

liquid  measures  X 0.859  367  (aprx.  subtr.  Vr)  =  dry  meas.  (log  1.934  1787) 

Th3  fluid  ounce  is  in  use  chiefly  by  apothecaries.  The  barrel  is  no  fixed 
unit,  and  no  value  of  a  barrel  has  ever  been  adopted  by  Congress;  in  the 
Customs  Service  and  Internal  Revenue  Bureau  every  barrel  is  gauged. 

Concerning  the  bushel  tho  Revised  Statutes  of  the  United  States,  under 
the  ascertainment  of  duties  on  grain,  chapter  6,  sec.  2919,  says:  "For 
the  purpose  of  estimating  the  duties  on  importations  of  grain,  the  num- 

t  Guillaume  in  n,  recent  book  on  the  International  Bureau  of  Weights 
and  Measures  states  that  this  Bureau  has  found  the  mass  of  a  cubic  deci- 
meter of  water  at  4°  C.  to  be  0.999  955  kg.  This  makes  a  liter  about  45 
parts  in  1000000.  greater  than  a  cubic  decimeter.  But  it  seems  that  no 
formal  adoption  of  any  value  has  yet  been  made. 


46  VOLUMES. 

ber  of  bushels  shall  be  ascertained  by  weight,  instead  of  by  measuring; 
and  sixty  pounds  of  wheat,  fifty-six  pounds  of  corn,  fifty-six  pounds  of 
rye,  forty-eight  pounds  of  barley,  thirty-two  pounds  of  oats,  sixty  pounds 
of  pease  and  forty-two  pounds  of  buckwheat,  avoirdupois  weight,  shall 
respectively  be  estimated  as  a  bushel." 

In  Great  Britain  the  liter  is  exactly  the  same  as  in  the  United  States. 
The  gallon,  which  in  this  country  is  often  called  the  Imperial  or  British 
gallon,  was  originally  defined  as  the  volume  of  "ten  imperial  pounds  of 
distilled  water,  weighed  in  air  against  brass  weights,  with  the  water  and 
the  air  at  the  temperature  of  62°  Fahrenheit,  the  barometer  being  at  30 
inches."  According  to  the  latest  computations  of  the  Standards  Office 
in  London,  the  British  or  Imperial  gallon  is  equal  to  4.545963  1  liters; 
this  value  was  made  legal  in  1896  t  m  Great  Britain,  and  is  the  one  used 
as  a  basis  throughout  these  tables,  even  for  computing  such  values  as 
the  gallon  in  cubic  inches ;  the  figures  are  therefore  quite  consistent  through- 
out, except  that  the  U.  S.  yard,  foot,  inch  are  used,  which  are  about  3 
parts  in  one  million  larger  than  the  British,  a  difference  too  small  to  be 
considered  in  any  but  the  most  refined  physical  measurements.  Owing 
to  this  difference,  the  U.  S.  cubic  yard,  cubic  foot,  etc.,  are  about  9  parts 
in  one  million  greater  than  the  British.  Hence  to  reduce  these  values 
in  those  tables  to  British  yards,  feet,  and  inches,  when  very  great  accuracy 
is  required,  subtract  9  parts  for  every  million  from  all  the  values  of  a  cubic 
yard,  cubic  foot,  and  cubic  inch  given  in  terms  of  other  units  than  yards, 
feet,  and  inches;  for  instance,  add  this  correction  .in  the  value  of  a  cb. 
foot  in  cb.  meters,  but  not  in  cb.  inches,  as  the  latter  is  the  same  for  both; 
or  add  9  parts  for  every  million  to  all  the  values  of  other  units  given  in 
terms  of  cubic  yards,  cubic  feet,  and  cubic  inches.  This  will  never  affect 
the  fifth  place  of  figures  by  more  than  one  unit,  and  generally  not  that 
much. 

In  Great  Britain  the  measures  of  capacities  (gallons,  quarts,  etc.)  are 
the  same  for  liquid  as  for  dry  materials.  The  British  measures  of  capacity 
differ  but  little  from  the  corresponding  dry  measures  in  the  United  States; 
accurately: 

Brit.  meas.  X  1.032  02  (aprx.  add  Mo)  =  dry  U.  S.  meas.  (log  0-013  6888) 
dry  U.S.  meas.  X  0.968  972  (aprx.  sub.  Mo)  =  Brit.  meas.  (log  1-9863112) 
The  British  measures  of  capacity  are  about  20%  greater  than  the  corre- 
sponding  liquid  measures  in  the  United  States;    accurately: 

Brit.  meas.  XI. 200  91  (aprx. add %}  =  liquid  U.S.  meas.  (log  0-079  5101) 
liquid  U.S.  meas.  X  0.832  602  4  (aprx.  i%2)  =  Brit.  meas.  (log  1.920  4899) 
The  U.  S.  and  the  British  apothecary  fluid  measures,  smaller  than  the 
pint,  differ  very  little  from  each  other,  almost  exactly  4%,  the  former 
being  the  greater;    for  conversion  add  4%  to  the  quantity  expressed  in 
the  U.  S.  measures,  or  subtract  4%  when  expressed  in  British  measures. 

VOLUMES.     Cubic  and  Capacity  Measures.     Usual. 

**  Accepted  by  the  National  Bureau  of  Standards. 

*    Checked  by  L.  A.  Fischer.  Asst.  Phys.  National  Bureau  of  Standards. 

Aprx.  means  within  2%. 

ap.  means  apothecary  measures:  cb.  means  cubic,  and  is  often  used  as  a 
suffix  instead  of  the  exponent  (3),  being  simpler  for  printing  and  type 
writing;  thus  cb.  cm.  for  cm.3,  or  cb.  ft.  for  ft.3 

1  cb.  centimeter  [cm3]  or  1  milliliter  [ml]:  Logarithm 

=  1  000.  cb.  millimeters 3-000  0000 

=      0.061  023  4*  cb.  inch.    Aprx.6-M00. 2-7854962 

=  0.01    deciliter 2-000  0000 

=   0.00211336    pint*  (liquid;  U.S.).    Aprx.  21  -s- 10  000.  .  3-3249742 

=   0.001  816  16*  pint  (dry;  U.S.).    Aprx.  %i -MOO 3-259  1529 

=   0.001  759  80    pint  (Brit.).    Aprx.  %-M  000 3-245  4641 

-    0.001  056  68*  quart  (liquid;   U.  S.).    Aprx.  2l/2-^10  000.  3-023  9442 

0.001    liter  or  cb.  decimeter 3-000  0000 

-0.000  908  078*  quart  (dry;  U.  S.).    Aprx.  9-5- 10  000 4-958  1229 

=  0.000  879  902    quart  (Brit.).    Aprx.  K-M  000 4-944  4341 

t  Formerly  the  legal  value  is  said  to  have  been  4.543  46  liters. 


VOLUMES.  47 

1  cb.  inch  [in3]  =  16  387.16*  cb.  mm.    Aprx.  MX  100  000.  ..  4-214  5038 

••               =  16.387  16*  cb.  cm.  or  ml.    Aprx.  2^X100..    1-2145038 

••  =      0.163  871  6*  deciliters.    Aprx.  Y& 

=  0.0346320*  pt.  (liq.;  U.S.).  Aprx.  %-*- 100. 

=  0.029  761  6*  pt.  (dry;  U.  S.).  Aprx.  3-^100. 

=     0.0288382    pt.  (Brit.).    Aprx.  %  +•  10 

•'               =  0.017  316  0*  qt.  (liq.; U.S.).     Aprx. %-s- 100. 

"               =  0.01638716    liter  or  cb.  dm.    Aprx.  Y&-r- 10.. 

"               =  0.014880  8*  qt.  (dry;  U.S.).    Aprx.  %•*-  100 

"  =      0.0144191     qt.  (Brit.).    Aprx.  ^-J- 10 2-1589379 

=  0.004  329  00*  gal.  (liq. ;  U.  S.).    Aprx.  %  -4-100  3-636  3880 

"               =  0.00360477    gal.  (Brit.).    Aprx.  fn -*- 100.  ..   3-5568779 

"  =0.000  578  704*  cb.  foot.    Aprx.  #  -r-  1  000 4-762  4563 

1  deciliter  [dl]  =  100.  cb.  centimeters 2-000  0000 

6.102  34*  cb.  inches.    Aprx.  6Ho 0-785  4962 


0.211  336*  pt.  (liq. ;  U.  S.).    Aprx.  21  H-  100. 

0.181  616*  pt.  (dry;  U.  S.).    Aprx.  %i 

0.175  980    pint  (Brit.).    Aprx.  %  + 10 

0.105  668*  qt.  (liq.;  U.  S.).    Aprx.  2^-5- 100. 
0.1    liter  or  cb.  decimeter, 


324  9742 
-259  1529 
-245  4641 
-023  9442 
-000  0000 


=   0.0908078*  quart  (dry;  U.S.).    Aprx.  Ml-. --  2-958  1229 

••  =   0.0879902    quart  (Brit.).    Aprx.  ]/$  + 10.  .  .  .  2-9444341 

=   0.026  417  0*  gal.  (liq.  U.  S.).    Aprx.  %-*- 100.  .  2-421  8842 

=   0.0219975    gal.  (Brit.).    Aprx.  22 -^  1  000.  .  .  2-3423741 

"  =0.003  531  45*  cb.  foot.    Aprx.%-^-1  000 3.547  9525 

1  pint  [pt]  (liquid;  U.  S.): 

=          .  473.179*  cb.  centimeters.    Aprx.  Hi  X  10  000 2-675  0258 

=  28.875*  cb.  inches.    Aprx.  ft  X  100 1-460  5220 

=  16.  fluid  ounces  (ap.  U.  S.) 1-204  1200 

=  4.731  79*  deciliters.    Aprx.  4% 0-675  0258 

=  4.  gills  (liquid;  U.  S.).  .  . '. 0-602  0600 

=         0.859  367*  pint  (dry;  U.  S.).    Aprx.  % 1-934  1787 

=     0.832  702  4    pint  (Brit.).    Aprx.  % 1.920  4899 

=  0.5    quart  (liquid;  U.  S.);  or  \i 1-698  9700 

=         0.473  179*  liter  or  cb.  decimeter.    Aprx.  Ml  X  10 1-675  0258 

=  0.125    gallon  (liquid;  U.S.):  or H 1-0969100 

=      0.016710  1*  cb.  foot.    Aprx.  M- 10.  . 2-2229783 

=   0.004  731  79*  hectoliter.    Aprx.  Hi  -*•  10 3-675  0258 

=  O-.OOO  618  891*  cb.  yard.    Aprx.  ^g -7-1  000 4-7916145 

=  0.000473  179*  cb.  meter  or  stere.    Aprx.Ki^-100 4-6750258 

1  pint  [pt]  (dry;  U.  S.): 

=  550.614*  cb.  centimeters.    Aprx.  550 2-740  8471 

=       33.600  312*  cb.  inches.    Aprx.^XlOO 1-5263433 

=  5.506  14*  deciliters.     Aprx.  5^  or  ^ 0-740  8471 

=  1.16365*  pints  (liquid;  U.S.).    Aprx.l^ory6 0-0658213 

=         0,968  972    pint  (Brit.).    Aprx.  subtract  ^0 1-986  3112 

=.        0.550  614*  liter  or  cb.  decimeter-    Aprx.  "^  10 .  1-740  8471 

=  0.5    quart  (dry;  U.S.);  or  ^ 1-6989700 

-  0.121  122    gallon  (Brit.).    Aprx.  %  •*•  10 1-083  2215 

=         0.062500*  peck  (U.S.);  or  ^10 2-7958800 

«     0.019  444  6*  cb.  foot.    Aprx.  2yn  -=- 100 2-2887996 

=         0.015  625*  bushel  (U.  S.).    Aprx.H<r^lOO 2-1938200 

==   0.005  506  14*  hectoliter.    Aprx.  ^-^  1  000 3-740  8471 

=  0.000  720  171*  cb.  yard.    Aprx.  fy  +  l  000 4-857  4358 

=  0.000  550614*  cb.  meter  or  stere.    Aprx.  ^-J- 10  000.  .  .  .  4-740  8471 

1  pintfpt]  (Brit.): 

568.245  39   cb.  centimeters.    Aprx.  #  X 1  000 2-754  5359 

=  34.6762   cb. inches.    Aprx.%xlO 1-5400321 

=  20.  fluid  ounces  (ap.  Brit.) 1-301  0300 

—  5.682  453  9   deciliters.    Aprx.  5% 0-754  5359 

-  4.  gills  (Brit.) 0-602  0600 

•=  1.20091   pints  (liquid;  U.S.).    Aprx.  1% 0-079  5101 

=  1.03202   pints  (dry;  U.S.).    Aprx.  1  Ho 0-0136888 

=         0.568  245  39  liter  or  cb.  decimeter.    Aprx.  ty 1-7545359 

=  0.5   quart  (Brit.);  or^ 1-698  9700 

=  0.125   gallon  (Brit.);  or  1A 1-0969100 

-  0.062  5   peck  (Brit.);  or^-r-10 2-795  8800 


48 


VOLUMES. 


1  pint  [pt]  (Brit.) . 

=           0.020  067  3  cb.  foot.    Aprx.  2  -*- 100 2-302  4884 

0.015  625  bushel  (Brit.).    Aprx.  1^7-*- 100 * 2-1938200 

=   0.005  682  453  9  hectoliter.    Aprx.  #•*•  100 3.754  5359 

=       0.000  743  232  cb.  yard.    Aprx.^-^  1  000 1.871  1246 

=  0.000  568  245  39  cb.  meter  or  stere.    Aprx.  ty  -f- 1  000 1-754  5359 

1  quart  [qt]  (liquid;  U.  S.). 

946.359*  cb.  centimeters.    Aprx.  %i  X  10  000 2-976  0558 

=           57.750  0*  cb.  inches.    Aprx.  1^X100 1-7615520 

=           9.463  59*  deciliters.    Aprx.  %i  X  100 0-976  0558 

=                        8.  gills  (liquid;  U.  S.) 0-903  0900 

=                       2.  pints  (liquid;  U.  S.) 0-301  0300 

=         0.946  359*  liter  or  cb.  decimeter.    Aprx.  subtr.  ^0- ...  1-976  0558 

=         0.859  367*  quart  (dry;  U.S.).    Aprx.  % 1-934  1787 

=     0.832  702  4    quart  (Brit.).    Aprx.  % 1.920  4899 

=                  0.25*  gallon  (liquid;  U.S.);  or  K 1-3979400 

=         0.208170    gallon  (Brit.).    Aprx.21-^-100 1-3184299 

=     0.033  420  1*  cb.  foot.    Aprx.  }/$*- 10 2-524  0083 

=   0.009  463  59*  hectoliter.    Aprx.  %i-*-10 3-976  0558 

=   0.001  237  78*  cb.  yard.    Aprx.  14 •*- 100 3-092  6445 

=  0.000  946  359*  cb.  meter  or  stere.    Aprx.  %i -i- 100 4-976  0558 

1  cb.  decimeter  [dm3]  =  1.  liter  which  see  for  values 0-000  0000 

1  liter  [1]=              1  000.  cb.  centimeters 3.000  0000 

"         =         61.0234*  cb.  inches.    Aprx.  60 1-7854962 

"         =                    10.  deciliters 1.000  0000 

=         2.113  36*  pints  (liquid;  U.S.).    Aprx.  2i^0 0-3249742 

"         =         1.816  16*  pints  (dry;  U.  S.).    Aprx.  20^ 0-259  1529 

=         1.75980    pints  (Brit.).    Aprx.  l%or% 0-2454641 

=         1.056  68*  quarts  (liquid;  U.  S.).    Aprx.  add  Ko-.  0-023  9442 

=                      1.  cb.  decimeter 0-000  0000 

=       0.908  078*  quart  (dry;  U.  S.).    Aprx.*Ho-  .  .  . 1-958  1229 

=       0.879  902    quart  (Brit.).    Aprx.  % , 1.944  4341 

=      0.264  170*  gallon  (liquid^  U.  S.).    Aprx.  %0 1-421  8842 

=      0.219975    gallon  (Brit.).    Aprx. 22^-100 ".342  3741 

=•      0.113510    peck(U.S.).    Aprx.^-r-lO 1-0550329 

=      0.109  988    peck  (Brit.).    Aprx.  11  -*- 100 1.041  3441 

=   0.0353145    cb.foot.    Aprx.  %  + 100 2-5479525 

=   0.0283774    bushel  (U.  S.).    Aprx.  2^-=- 10 2-4529729 

=   0.027  496  9    bushel  (Brit.).    Aprx.  114  -*-100 2-439  2841 

=                0.01    hectoliter 2-000  0000 

=  0.00130794    cb.yard.    Aprx.  13 -r- 10  000 3-1165887 

=              0.001    cb.  meter  or  stere 3-000  0000 

t  quart  [qt]  (dry;  U.  S.): 

=         1101.23    cb.  centimeters.    Aprx.  1  100 3-0418771 

=     67.200625    cb.  inches.    Aprx.  ^X 100 1-8273733 

=         11.0123    deciliters.    Aprx.  11 1-0418771 

2.  pints  (dry;  U.S.) 0-301  0300 

=         1.16365    quarts  (liquid;  U.S.).    Aprx.addH 0-0658213 

=         1.10123    liters  or  cb.  decimeters.    Aprx.  iMo 0-0418771 

=       0.968  972    quart  (Brit.).    Aprx.  subtract  Mo 1-986  3112 

=       0.242243    gallon  (Brit.).    Aprx.  24-h  100 1-3842512 

=              0.125*  peck  (U.  S.);  or  l/8 1-096  9100 

=   0.038  889  3*  cb.  foot.    Aprx.  He 2-589  8296 

=       0.031  250*  bushel  (U.S.).    Aprx.  31 -4- 1  000 2-4948500 

=   0.011  012  3    hectoliter.    Aprx.  11-^-1  000 2-041  8771 

=  0.001  44034*  cb.yard.    Aprx. ^-5- 100 3-1584658 

=  0.001  101  23    cb.  meter  or  stere.    Aprx.  11 -f- 10  000 3-041  8771 

1  quart  [qt]  (Brit.): 

=        1136.490  8  cb.  centimeters.    Aprx.  %  X 1  000 3-0555659 

=             69.352  5  cb.  inches.    Aprx.  70 1-841  0621 

=         11.364908  deciliters.    Aprx. 8^x10 1-0555659 

=                          8.  gills  (Brit.) 0-903  0900 

=                         2.  pints  (Brit.) 0-301  0300 

=             1.200  91   quarts  (liquid;  U.  S.).    Aprx.  \% 0-079  5101 

=        1.136  490  8  liters  or  cb.  decimeters.    Aprx.  add^ 0-055  53J9 

=             1.032  02  quarts  (dry;  U.  S.).    Aprx.  l^o 0-013  ".OC8 

*          0.300227  gallon  (liquid;  U.S.).    Aprx.^o 1-4774501 


VOLUMES. 


49 


1  quart  [qtl  (Brit.)* 

0.25  gallon  (Brit.);  or% 1-397  9400 

=  0.125  peck  (Brit.) ;  or  H 1-096  9100 

=       0.040  134  6  cb.  foot.    Aprx.  4 +  100 2-603  5184 

=  0.031  250  bushel  (Brit.).    Aprx.  31  -4- 1  000 1-4948500 

=   0.011  364  908  hectoliter.     Aprx.  %-s-lOO 2-055  5659 

=      0.001  486  46  cb.  yard.     Aprx.  % 4-1  000 g.172  1546 

=  0.001  136  490  8  cb.  meter  or  stere.    Aprx.  %  -*- 1  000 §.055  5659 

1  gallon  [gal]  (liquid;  U.  S.): 

=         3785.43*    cb.  centimeters.    Aprx.  H  X  10  000 3-5781158 

=  2S1.**  cb.  inches.    Aprx.  %  X 100 2-363  6120 

=         37.854  3*    deciliters.    Aprx.  */*  X  100 1-578  1158 

=  32.     gills  (liquid;  U.  S.) 1-505  1500 

=  8.     pints  (liquid;  U.S.) 0-9030900 

=  4.     quarts  (liquid;  U.  S.) 0-602  0600 

=-         3.78543*    liters  or  cb.  decimeters.    Aprx.  HX 10.  ...  0-578  1158 

=   0.832  702  4      gallon  (Brit.).    Aprx.  % 1.920  4899 

=       0.133681*    cb.  foot.    Aprx.%-!-10 '1-1260683 

=   0.037  854  3*    hectoliter.    Aprx.  %  X  10 2-578  1158 

=  0.00495113*    cb.yard.    Aprx.^-f-100 3-6947045 

=  0.00378543*    cb.  meter  or  stere.    Aprx.^^-100 3-5781158 

I  Imperial  gallon  [see  gallon  (Brit.)]. 
1  gallon  [gal]  (Brit.)- 

4  545.963  1   cb.  centimeters.    Aprx.  %X  1  000 3-657  6259 

277.410  cb.  inches.    Aprx.  ^XlOO 2-443  1221 

32.  gills  (Brit.) L505  1500 

8.  pints  (Brit.) 0-903  0900 

=     4.545  963  1   liters  or  cb.  decimeters.    Aprx.  4^ 0-657  6259 

4.  quarts  (Brit.) 0-602  0600 

1.200  91   gallons  (liquid;  U.  S.).    Aprx.  1% 0-079  5101 

0.160  538  cb.foot.    Aprx.  %-5- 10 1-205  5784 

0.125  bushel  (Brit.);  or^g 1-096  9100 

=     0.045459631   hectoliter.    Aprx.  %-s- 100 2-6576259 

=       0.00594586  cb.yard.    Aprx.  6-=-l  000 3-7742146 

=  0.0045459631   cb.  meter  or  stere.    Aprx.  %-f- 1  000 3-6576259 

peck  [pk]  (U.  S.)  =         8  809.82*  cb.  cms.    Aprx.  %  X  10  000.  .   3-944  9671 
=       537.605  0*  cb.  ins.    Aprx.  «/u  X  1  000.  .  .   2-730  4633 

=  16.  pints  (dry;  U.S.) 1-2041200 

=         8.809  82*  lit's  or  cb.  dms.  Aprx.  V%  X  10  0-944  9671 

=  8.  quarts  (dry;  U.  S.) 0-903  0900 

=  1.93794  gals.  (Brit.).  Aprx.  21/11-  ...  0-287  3412 
=  0.968  972  pk.  (Brit.).  Aprx.  subt.  ^0-  -  1-986  3113 
=  0.311  114*  cb.  foot.  Aprx.  31  + 100.  .  .  .  1.492  919« 

=  0.25    bushel  ( U.  S.)  or  M 1-3979400 

—   0.088  098  2*  hectoliter.    Aprx.  %-f- 10 2-944  9671 

=   0.011  522  7*  cb.yard.    Aprx.  %-s- 100 2-061  555* 

=  0.00880982*  cb.  meter.    Aprx.  %-*•  100...  .   3-9449671 

t  peck  [pk]  (Brit.)=         9  091.926  2  cb.  cms.    Aprx.  9  000 3-958  6559 

=  554.820  cb.  ins.    Aprx.  ^X  100. ..   2-744  1521 

=  16.  pints  (Brit.) 1-2041200 

=         9.091  926  2  liters  or  cb.  dms.    Aprx.  9..  0-958  6559 

=  8.  quarts  (Brit.) 0-903  0900 

=  2.  gallons  (Brit.) 0-301  0300 

=  1.03202  pecks  (U.S.).    Aprx.  IHo--  0-013  6888 

=  0.321  076  cb.foot.    Aprx.  32-^100  .  .   1-506  6084 

=  0.25  bushel  (Brit.)  or  M 1-397  9400 

=  0.090  919  262  hectoliter.  Aprx.  9-*- 100..  2-9586559 
=  0.011  891  7  cb.  yard.  Aprx.  12-J-l  000  2-075  2446 
=  0.0090919262  cb.  meter.  Aprx.  9-^1  000.  3-9586559 

t  cb.  foot  [ft3]=       28  317.0*  cb.  cms.    Aprx.  %  X  100  000 4-452  0475 

=  1  728.*^.  inches.    Aprx.  %  X  1  000 3-237  5437 

=       59.8442     pints  (liq.;  U.S.).    Aprx.  60 1-7770217 

=     51.42809     pints  (dry;  U.S.).    Aprx.  51 1-7112004 

=       49.8324     pints(Brit.).    Aprx.  50 1-6975116 

—       29.9221      quarts  (liq.;  U.S.).    Aprx.  30 1-4759917 

=       28.3170*   liters  or  cb.  dms.  Aprx.  %X  100...   1-4520475 
«     25.71405*   quarts  (dry;  U.S.).    Aprx.  26 1-4101704 


50 


VOLUMES. 


1  cb.  foot[fl3]  =       24.9162    quarts  (Brit.).    Aprx.  25 1-3964818 

=       7.48052*  gals,  (liq.;  U.S.).    Aprx.  ^X  10.  ..   0-873  9317 
=       6.42851*  gallons  (dry;  U.S.).    Aprx.  6^-  .   0-808  1104 

=       6.229  05    gallons  (Brit.).    Aprx.  6J4 0-794  4216 

=       3.21426*  pecks  (U.S.).    Aprx.  3*4  ... 0-5070804 

=       3.11452    pecks  (Brit.).    Aprx.  3H 04933916 

=     0.803564*  bushel  (U.S.).    Aprx.  % 1-905  0^04 

=     0.778  630    bushel  (Brit.).    Aprx.  % I  891  3316 

=     0.283  170*  hectoliter.    Aprx.  % .-  1-452  0475 

=  0.037  037  0*  cb.  yard.    Aprx.  ^10 2-5686362 

=  0.0283170*  cb.  meter  or  stere.    Aprx.  %-*- 10.  .   2-4520475 
1  bushel  [bu]  (U.  S.)' 

=         35  239.28*    cb.  centimeters.    Aprx.  %X  10  000 4-547  0271 

=  3  150.4300**  cb.  inches.    Aprx.  1%  X  1  000 3-3325233 

64.     pints  (dry;  U.S.) 1-806  1800 

=         35.239  28*    liters  or  cb.  decimeters.    Aprx.  35 1-547  0271 

=  32.     quarts  (dry;  U.  S.) 1-505  1500 

=  7.751  78      gallons  (Brit.).    Aprx.  7M 0-889  4012 

=  4.     pecks  (U.  S.) 0-602  0600 

=  1.244  46*    cb.  feet.    Aprx.  \\i 0-094  9796 

=         0.968  972      bushel  (Brit.).    Aprx.  subtr.  Y^ 1.986  3112 

=     0.352  392  8*    hectoliter.    Aprx.  %-5- 10 1-547  0271 

=  0.125      quarter  ( U.  S.)  or  1A 1-0969100 

=     0.046  091  0*    cb.  yard.    Aprx.  %i  -=- 10 2-663  6158 

=   0.035  239  28*    cb.  meter  or  stere.    Aprx.  %  +  100 2-547  0271 

1  bushel  [bul  (Brit.): 

=       36  367.704  8  cb.  centimeters.    Aprx.  Vn  X  100  000 4-560  7159 

=  2  219.28  cb.  inches.    Aprx.  %  x  10  000. .  . 3  346  2121 

=  64.  pints  (Brit.) 1-806  1800 

=       36.367  704  8  liters  or  cb.  decimeters.    Aprx.  4/u  X  100.   1-560  7159 

=  32.  quarts  (Brit.) 1-505  1500 

=  8.  gallons  (Brit.) 0-903  0900 

=  4.  pecks  (Brit.) 0  602  0600 

=  1.284  31   cb.  feet.    Aprx.  % 0-108  6684 

=  1.032  02  bushels  (U.  S.).    Aprx.  addVao 0  013  6888 

=  0.125   quarter  (Brit.)  or  H 1-096  9100 

=         0.047  566  9  cb.  yard.    Aprx.  48  + 1  000 2-6773046 

=  00363677048  cb.  meter  or  stere.    Aprx. Vn^- 10 2-5607159 

1  hectoliter  [hl]  =   6  102.34*  cb.  inches.    Aprx.  6  000 3-785  4962 

=     211.336    pints  (liq.;  U.S.).    Aprx.  210 2-3249742 

=     181.616*  pts.  (dry;  U.  S.).  Aprx.  *KX  100..   2-259  1529 

=     175.980    pints  (Brit.).    Aprx.  %  X  100 2-245  4641 

=     105.668    quarts  (liquid;  U.S.)    Aprx.  106.    2-0239442 

"  =  100.  liters  or  cb.  decimeter 2-000  OOQO 

=   90.8078    quarts  (dry;  U.S.).    Aprx.  90 1-9581229 

=   87.9902    quarts  (Brit.).    Aprx.  7AX  100  ...  .    19444341 
=   26.4170    gals,  (liq.;  U.S.).    Aprx.%XlO..   1-4218842 

=   21.9975    gallons  (Brit.).    Aprx.  22 1-3423741 

=    11.3510    pecks(U.S.).    Aprx.^XlO 10550329 

=    10.998  8    pecks  (Brit.).    Aprx.  11 1-041  3441 

=   3.531  45    cb.  feet.    Aprx.  3^  or  % 0-547  9525 

=   2.837  74    bushels  (U.  S.).    Aprx.  %  X  10 0-452  9729 

=   2.74969    bushels  (Brit.).    Aprx.  ^ 0-4392841 

=  0.130794    cb.yard.    Aprx.  13-^  100 1-1165887 

=  0.1    cb.  meter  or  stere 1-000  0000 

1  «b.  yard  [yd3l=-       46  656.  cb.  inches.    Aprx.  1%  X  10  000 4-668  9075 

=    161579  pints  (liq.;  U.S.).  Aprx.  %X  1000.  3-2083855 
=    1388.56   pints  (dry;  U.S.).  Aprx.  %X  1000  3-1425642 

=    1345.47   pints  (Brit.).    Aprx.fsXlOOO 3-1288754 

=  807.896  quarts  (liquid;  U.S.).  Aprx.  800..  2-9073555 
=  764.559  liters  or  cb.  dms.  Aprx.  M  X  1  000 
•=  694.279  quarts  (dry;  U.S.).  Aprx.  700... 
=  672.737  quarts  (Brit.).  Aprx.  %  X  1  000.. . 
=  201.974  gallons  (liq.;  U.S.).  Aprx.  200.  .. 
-  =  168.184  gallons  (Brit.).  Aprx.  KX  1000. 

=   86.7849  pecks  (U.S.).    Aprx.  ^X  100 

=   84.092  1   pecks  (Brit.).    Aprx.  %  X  100 


2-883  4113 
2-841  5342 
2-827  8454 
2-305  2955 
2-225  7854 
1-938  4442 
1-924  7554 


VOLUMES. 


51 


I  cb.  yard  [yd3]  =.  27.  cb.  feet.    Aprx.  %  X  10 1.431  3638 

=   21.6962  bushels  (U.S.).    Aprx.  11/5  X  10.  ...    L336  3842 

=   21.0230  bushels  (Brit.).    Aprx.  21 1-3226954 

=  0.764559  cb.  meter  or  stere.    Aprx.  % 1-8834113 

L  cb.  meter  [m3]  or  stere  [s]: 

=  61  023.4  cb.  inches.    Aprx.  60  000 4.735  4962 

-2  113.36  pints  (liquid;  U.  S.).    Aprx.  2  100 3.324  9742 

=  1  816.15  pints  (dry;  U.S.).    Aprx.  ^X  1000 3-2591529 

=  1  759.80  pints  (Brit.).    Aprx.  %  X  1  000 3.245  4641 

=  1056.68  quarts  (liquid;  U.S.).    Aprx.  2KX  100 3-0239442 

=       1  000.  liters  or  cb.  decimeters 3-000  0000 

=  908.078  quarts  (dry;  U.S.).    Aprx.  900 2-9581229 

=  879.902  quarts  (Brit.).    Aprx.  Jf  X 1  000 2-944  4341 

=  264.170  gallons  (liquid;  U.  S.).    Aprx.  %X  100 2-421  8842 

=  219.975   gallons  (Brit.).    Aprx.  220 2-342  3741 

=  113.510  peek? (U.S.).    Aprx.  %X  100 2-0550329 

=  109.988  pecks  (Brit.).    Aprx.  110 2-041  3441 

--=  35.314  5  cb.  feet.    Aprx.  %  X  10 1.547  9525 

=  28.3774  bushels  (U.S.).    Aprx.  %  X  100. 1-4529729 

=  27.4969  bushels  (Brit.).    Aprx.  ^X  10 1-4392841 

10.  hectoliters 1-000  0000 

=  1.307  94  cb.  yards.    Aprx.  1^ 0.116  5887 


Conversion  Tables  for  Volumes.     Cubical;    Capacity. 


Cb.ins.  = 
Cb.cms  = 
Cb.  ft.  = 
Cb.met= 
Cb.yds  = 
Fl.drs.= 

cb.cms 

. 

3b.  ins..  . 

fluid 
drams. 

cb.  met'  s 

cb.feet 

cb.  yds 
1 

cb.cms 
m.  lit'r 

cb.met's 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

16.387 
32.774 
49.161 
55.549 
81.936 
98.323 
114.71 
131.10 
147.48 
183.87 

3.061  023 
3.12205 
3.18307 
3.24409 
3.305  12 
3.366  14 
3.427  16 
3.488  19 
3.54921 
3.610  23 

0.028  317 
0.056  634 
0.084951 
0.11327 
0.14158 
0.16990 
0.19822 
3.22654 
0.254  85 
0.283  17 

35.314 
70.629 
105.94 
141.26 
176.57 
211.89 
247.20 
282.52 
317.83 
353.14 

0.764  56 
1.5291 
2.2937 
3.0582 
3.822  8 
4.5874 
5.351  9 
6.1165 
6.881  0 
76456 

»  1.307  9 
2.6159 
3.9238 
5.231  8 
6.539  7 
7.847  6 
9.1556 
10.463 
11.771 
13.079 

3.696  7 
7.393  4 
11.090 

14.787 
18.484 

22.180 
25.877 
29.574 
33.270 
36.967 

3.270  51 
3.541  02 
3.81153 
1.0820 
1.3526 
1.6231 
1.8936 
2.164  1 
2.434  6 
2.705  1 

Fl.  oz.     = 
Cb.  cms.= 
Quarts   =« 
Liters     = 
Gallons.  = 
Bushels  = 
Hectol  t  = 

cb.cmg 

fl.  ounce? 

liters 

quarts 


liters 

gallons 

hectol'  ts 

bush'la 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

29.574 
59.147 
88.721 
118.29 
147.87 
177.44 
207.02 
236.59 
266.16 
295.74 

0.033814 
0.067  628 
0.10144 
0.13526 
0.16907 
0.20288 
0.236  70 
0.27051 
0.304  32 
0.338  14 

0.946  36 
1.8927 
2.839  1 
3.785  4 
4.731  8 
5.6782 
6.624  5 
7.5709 
8.5172 
9.4636 

1.0567 
2.1134 
3.1700 
4.226  7 
5.2834 

6.340  1 
7.396  8 
8.4535 
9.510  1 
10.567 

3.785  4 
7.570  9 
11.356 
15.142 
18.927 
22.713 
26.498 
30.283 
34.069 
37.854 

3.264  17 
0.528  34 
0.79251 
1.0567 
1.3209 
1.5850 
1.8492 
2.1134 
2.377  5 
2.641  7 

0.352  39 
0.704  79 
1.0572 
1.4096 
1.7620 
2.1144 
2.466  7 
2.819  1 
3.1715 
3.523  9 

2.837  7 
5.675  5 
8.5132 
11.351 
14.189 
17.026 
19.864 
22.702 
25.540 
28.377 

52  VOLUMES. 

VOLUMES.     Cubic  and  Capacity  Measures  (continued). 
Unusual,  Special  Trade,  or  Obsolete. 

ap.  means   apothecary  measures;   av.  means  avoirdupois  weights";   aprx. 

means  approximately. 

1  molecule.  There  are  about  a  million,  million,  million  molecules  in 
a  cb.  millimeter  of  air  (Woodward). 

I  cubic  millimeter  [mm3]  =  0.001  milliliter  or  cb.  centimeter. 

1  minim  or  drop  (ap.  Brit.)  =  59.  192  2  cb.  millimeters  =  ^0  fluid  scruple 
(ap.  Brit.)  =  0.059  1922  milliliter  or  cb.  centimeter  =  ^0  fluid  dram  (ap. 
Brit.).  1  milliliter  or  cb.  centimeter  =  16.894  1  minims  or  drops  (ap.  Brit.). 

1  minim  or  drop  (ap.  U.  S.)  =  61.612  cb.  millimeters  =  0.061  612  milli- 
liter or  cb.  centimeter  =  )^o  fluid  dram  (ap.  U.  S.).  1  milliliter  or  cb.  centi- 
meter =  16.230  6  minims  or  drops  (ap.  U.  S.). 

1  milliliter  [ml]  =  1  cb.  centimeter,  which  see  above  in  other  table; 
it  equals  0.001  liter. 

1  fluid  scruple  (ap.  Brit.)  =  20.  minims  (ap.  Brit.  )  =  1.183  845  milliliters 
or  cb.  centimeters  =  ^  fluid  dram  (ap.  Brit.)  =  ^4  fluid  ounce  (ap.  Brit). 
1  milliliter  or  cb.  centimeter  =  0.844  705  fluid  scruple  (ap.  Brit.). 

1  fluid  drachm  same  as  fluid  dram. 

1  fluid  dram  (ap.  Brit.)  =  60.  minims  (ap.  Brit.  )  =  3.  551  534  milliliters 
or  cb.  centimeters  =  3.  fluid  scriiples  (ap.  Brit.)  =  0.216  73  cb.  inch  =  >£  fluid 
ounce  (ap.  Brit.)=*46o  pint  (Brit.).  1  cb.  centimeter  =  0.281  57  fluid 
dram  (ap.  Brit.). 

1  fluid  dram  (ap.  U.  S.)  =  60.  minims  (ap.  U.  S.)  =  0.225  59  (aprx.  %) 
cb.  inch  =  ^8  fluid  ounce  (ap.  U.  S.)=Vi28  pint  (liquid;  U.  S.)  =  3.69671 
milliliter  or  cb.  centimeter.  1  cb.  centimeter  =  0.270  51  fluid  dram  (ap. 
U.  S.). 


.     .. 

1  centiliter  [cl]  =  10.  cb.  centimeters  =  0.610  234  cb.  inch  =  0.01  liter. 
1  fluid   ounce  [fl  ozl  (ap.  Brit.)  =  480.  minims    (ap.  Brit.)  =  28.412  27 
cb.  centimeters  or  milliliters  =  24.  fluid  scruples  (ap.  Brit.  )  =  8.  fluid  drams 


.  .  .          .          . 

(ap.  Brit.)  =  1.733  81  cb.  mches  =  0.960  73  fluid  ounce  (ap.  U.  S.)  =  ^0  pint 
(Brit.)  =  0.028  412  27  liter.     1  liter  =  35.  196  fluid  ounces  (ap.  Brit.). 

1  fluid  ounce  [fl  oz]  (ap.  U.  S.)  =  480.  minims  (ap.  U.  S.)  =  29.573  7  cb. 
centimeters  =  8.  fluid  drams  (ap.  U.  S.)  =  1.804  69  (aprx.  %)  cb.  inches  = 
1.04088  fluid  ounces  (ap.  Brit.)  =  yi6  pint  (liquid;  U.  S.)  =  0.029  573  7 
liter.  1  liter  =  33.813  8  fluid  ounces  (ap.  U.  S.). 

1  gill  [gi]  (liquid;  U.  S.)  =  118.295  cb.  centimeters  =  7.218  750  cb.  inches 
=  1.18295  deciliters  =  0.832  7024  gill  (Brit.)  =  M  pint  (liquid;  U.  S.)  = 
Y%  quart  (liquid;  U  S.)  =  0.118  295  liter.  1  cb.  inch  =  0.138  528  gill  (liquid; 
U.  S.).  1  liter  =  8.453  44  gills  (liquid;  U.  S.). 

1  gill  [gi]  (Brit.)  =  142.061  35  cb.  centimeters  =  8.669  05  cb.  inches  = 
1.4206135  deciliters  =1.200  91  gills  (liquid;  U.  S.)  =  M  pint  (Brit.)  =  J/g 
quart  (Brit.)  =  0.142  061  35  liter.  1  cb.  inch  =  0.1  15  352  8  gill  (Brit.). 
1  liter  =  7.039  20  gills  (Brit.). 

1  pint[pt]  (ap.  U.  S.)  =  128.  fluid  drams  (ap.  U.  S.)  =  16.  fluid  ounces 
(ap.  U.  S.)  =  0.473  179  liter  =  same  as  liquid  U.  S.  pint. 

1  pintfpt]  (ap.  Brit.)  =  480.  fluid  scruples  (ap.  Brit.)  =  160.  fluid  drams 
(ap.  Brit.)  =  20.  fluid  ounces  (ap.  Brit.)  =  0.568  245  4  liter  =  same  as  ordi- 
nary Brit.  pint. 

1  inillistere  =  1  cb.  decimeter  =1  liter. 

1  pottle  (Brit.)  =  H  gallon  Brit. 

1  board  foot  =144.  cb.  mches  =  M2  cb.  foot. 

1  gallon  [gal]  (ap.  U.  S.),  same  as  ordinary  liquid  U.  S.  gallon  of  231. 
cb.  inches. 

1  gallon  [gal.]  (wine;  old  Brit.),  same  as  ordinary  liquid,  U.  S.  gallon 
of  231.  cb.  inches. 

1  gallon  (dry;  U.  S.)  (obsolete;  term  "^  peck"  used  instead)  = 
268.8025  cb.  inches  =  8.  pints  (dry;  U.  S.)  =  4.40491  liters  =  4.  quarts 
(dry;  U.  S.)  =  H  peck  (U.  S.). 

1  gallon  [gal]  (ap.  Brit.),  same  as  ordinary  Brit,  or  Imperial  gallon. 

1  lieer  gallon  (obsolete)  =  282.  cb.  inches  =  8.  beer  pints  =  4.  beev  quarts. 

1  decaliter  or  dekaliter  [dkl]  =  10.  liters  =1  centistere. 

1  centistere  =  10.  cb.  decimeters  or  liters  =1  decaliter  =  ^ioo  stere  or 
cb.  meter. 


VOLUMES.  53 

1  gram-molecule  of  any  gas  at  0°  C.  and  760  mm.  pressure  has  a  vol- 
ume of  22  380.  cb.  centimeters. 

1  foot  (solid,  timber)  =  1  cb   ft. 

1  solid  foot=l  cb.  ft. 

1  hektoliter  [hi],  same  as  hectoliter. 

1  decistere  [ds]=100.  cb.  decimeters  or  liters  =1  hectoliter  =  Mo  stere 
or  cb.  meter. 

1  firkin  =  9.  gallons  (liquid;  U.  S.)  =  7.4943  gallons  (Brit.)  =  34. 068  9 
liters  =  %  barrel.  Of  butter  =  56.  pounds  (av.). 

1  Winchester  bushel  [bu],  same  as  ordinary  bushel  in  U.  S.  =  2  150.42 
rb.  inches. 

1  struck  bushel,  same  as  ordinary  bushel  in  U.  S.  =  2  150.42  cb.  inches. 

1  heaped  bushel  =  1*4  struck  bushels  or  1M  ordinary  bushels  (U.  S.). 

1  bushel  =  60  pounds  of  wheat  =  56  pounds  of  corn  or  rye  =  48  pounds 
of  barley  =  32  pounds  of  oats  =  60  pounds  of  peas  =  42  pounds  of  buck- 
wheat. (U.  S.  Customs;  legal.) 

1  coomb  (Brit. )  =  16.  pecks  (Brit.)  =  4. bushels  (Brit. )=  1.454  708  hecto- 
liters =  H  quarter  (Brit.). 

1  barrel  [bbl],  has  no  legal  or  fixed  value ;  it  varies  between  30  and  43 
gallons  (liquid;  U.  S.).  Barrels  should  therefore  always  be  gauged.  The 
following  are  the  most  usual  values,  about  in  the  probable  order  of  im- 
portance: 

1  barrel  (liquid;  U.  S.)  =  31^  gallons  (liquid;  U.  S.)  =  4.211  0  cb.  feet  = 
1.192  4  hectoliters  =  ^  hogshead. 

1  barrel  (wine  and  brandy;  Brit.)  =  31J/£  gallons  (Brit. )  =  1.431  98 
hectoliters.  Also  given  as  36  gallons  (Brit.). 

1  barrel  (liquid;   U.  S.)  =  31.  gallons  (liquid;  U.  S.)  =  1.173  5  hectoliters. 

1  barrel  (refined  oil)  =  42.  gallons  (liquid;  U.  S.;  used  by  Standard 
Oil  Co.). 

1  barrel  (flour;  U.  S.)  =  3.  bushels  (U.  S.);  also  given  as  3.75  cb.  feet; 
"  legal"  (?)  weight  given  as  196  pounds  (av.). 

1  barrel  (dry;  U.  S.)  =  21^  bushels  (U.  S.)  =  26.756  cb.  feet  =  7.576  45 
hectoliters. 

1  barrel  (beer;  U.  S.)  =  36.  beer  gallons  of  282  cb.  inches  =  1.663  6 
hectoliters. 

1  barrel  (liquid;   Penna.)  =  32.  gallons  (liquid;   U.  S.). 

1  sack  (coal;    Brit.)  =  3  bushels  (Brit.)  =1/12  chaldron. 

1  decistere  =  1  hectoliter  =  0.1  stere  or  cb.  meter. 

1  tierce  (liquid;  U.  8.) -42.  gallons  (liquid;  U.  S.)  =  1.5899  hecto- 
liters =  %  hogshead  (U.  S.). 

1  tierce  (Brit.)  =  42.  gallons  (Brit.)  =  1.909  30  hectoliters  =  ^  hogs- 
head (Brit.). 

1  hogshead  [hhd]  (beer;  U.  S.  obsolete)  =  54  beer  gallons  of  282  cb. 
inches. 

1  hogshead  [hhd]  (liquid;  U.  S.)  =  63.  gallons  (liquid;  U.  S.)  =  2.  bar- 
rels of  31H  gallons  (liquid;  U.  S.)  =  2.384  8  hectoliters  =1%  tierces  (liquid; 
U.S.). 

1  hogshead  [hhd]  (Brit.)  =  63.  gallons  (Brit.)  =  2.863  96  hectoliters. 

1  quarter  [qr]  (dry;  U.  S.)  =  8.  bushels  (U.  S.)  =  2.819  1  hectoliters  =  M 
(aprx.)  tun  (Brit.). 

1  quarter  [qr]  (Brit.)  =  8.  bushels  (Brit.)  =  2.909  416  hectoliters  =  2. 
coombs  (Brit.)  =  K  (aprx.)  tun  (Brit.)  =  1/10  last  (Brit.). 

1  puncheon  (liquid;  U.  S.)  =  84.  gallons  (liquid;  U.  S.)  =  3.17976  hec- 
toliters =2.  tierces  (U.  S.). 

1  puncheon  (Brit.)  =  84.  gallons  (Brit.)  =  3.818  61  hectoliters  =  2. 
tierces  (Brit.). 

1  pipe  or  butt  (liquid;  U.  S.)  =  126.  gallons  (liquid;  U.  S.)  =  4.7696 
hectoliters  =  2.  hogsheads  (U.  S.). 

1  pipe  or  butt  (Brit.)  =  126.  gallons  (Brit.)  =  5.727  91  hectoliters  =  2. 
hogsheads  (Brit.). 

1  cord  foot  (wood)  =  4X4X1  foot  =  16.  cb.  feet  =  0.453  07  cb.  meter  or 
stere  =  J/g  cord. 

1  tun  (liquid;  U.  S.)  =  2f>2.  gallons  (liquid;  U.  S.)  =  9.539  3  hectoliters  = 
6.  tierces  (liquid;  U.  S.)=4.  hogsheads  (U.  S.)  =  3.  puncheons  (U.  S.)  =  2. 
pipes  or  butts. 

1  tun  (Brit.)  =  252.  gallons  (Brit.)  =  11.455  8  hectoliters  =  6.  tierces 
(Brit.)  =4.  hogsheads  (Brit.)  =  3.  puncheons  (Brit.)  =  2.  pipes  or  butts 
(Brit.). 


54  VOLUMES. 

1  perch  (masonry)  =  16^X1^X1  foot  =  24%  cb.  feet  (generally  25  cb. 
feet,  sometimes  22  cb.  feet)  =  0.916  67  cb.  yard  =0.700  85  stere  or  cb.  meter. 

1  solid  yard  =  l  cb.  yard. 

1  stere  [s]=  1  cb.  meter,  which  see  above  in  other  table. 

1  kiloliter  [kl]  =  1  000.  liters  =10.  hectoliters  =  1  cb.  meter  or  stere. 

1  gross  ton  (2240  pounds  av.)  displacement  of  water  =  35. 881  3  cb.  feet  = 
1.328  93  cb.  yards  =  1.016  05  cb.  meters. 

1  shipping  ton  (for  cargo;  U.  S.)  =  40.  cb.  feet  =  32.1426  bushels 
(U.  S.)  =  31.145  2  bushels  (Brit.)  =  11.326  8  hectoliters  =  1.132  68  cb.  meters. 

1  shipping  ton  (for  cargo;  Brit.)=42.  cb.  feet  =  33.7497  bushels 
(U.  S.)  =  32.702  5  bushels  (Brit.)  =  11. 893  1  hectoliters  =  1.189  31  cb.  meters. 

1  chaldron  (dry;  U.  S.)  =  44.8006  cb.  feet  =  36.  bushels  (U.  S.)  = 
12.686  1  hectoliters. 

1  chaldron  (coal;  Brit.)  =  12.  sacks-=36.  bushels  (Brit.)  =  58.658  cb. 
feet;  weight  3  136.  pounds  (av.). 

1  chaldron  (Canada)  =  58. 64  cb.  feet  or  about  45  bushels  (Brit.);  stated 
also  as  25.64  cb.  feet  or  about  20  bushels  (Brit.). 

1  chaldron  (Newcastle)  weight  5  936.  pounds  (presumably  coal). 

1  wey  (Brit.)  =  51. 372  4  cb.  feet  =  40.  bushels  (Brit.)  =  5.  quarters 
(Brit.)=^  last  (Brit.). 

1  register  ton  (shipping;  for  whole  vessels)  =  100.  cb.  feet  =  2. 831  7 
cb.  meters. 

1  last  (Brit.)  =  102.745  cb.  feet  =80.  bushels  (Brit.)  =  10.  quarters  (Brit.) 
=  2.  weys  (Brit.). 

1  cord  [c]  (wood)  =  4X4X8  feet  =  128.  cb.  feet  =  8.  cord  feet  =  3.624  58 
cb.  meters  or  stere.  1  stere  or  cb.  meter  =  0.275  894  cord. 

1  toise  (Canada)  =  261^  cb.  ft.  =  9.685  18  cb.  yds.  =  7.404  90  cb.  meters. 

1  rod  (brickwork;  Brit.)  =  16>£  feet  square X  14.  inches  =  272*4  sq.  feet 
of  14.  inch  wall;  conventionally  272.  sq.  feet  of  14.  inch  wall  =  317M  cb.  feet 
=  8.985  9  cb.  meters. 

1  rod  (engineering  works;  Brit.)  =  306.  cb.  ft.  =  HM  cb.  yards  =  8.665 
cb.  meters. 

1  myrialiter  or  myrioliter=  10  000.  liters  =100.  hectoliters  =  10.  cb. 
meters  or  steres=l  decastere. 

1  decastere  or  dekastere  [dks]=100.  hectoliters  =  10.  cb.  meters  or 
steres  =  1  myrialiter. 

1  hectostere  or  hektostere  [hks]=100.  steres  or  cb.  meters. 

1  acre-foot  (irrigation)  =  325  851.  gallons  =  43  560.  cb.  feet  =  1  613.33 
cb.  yards  =  1  233.49  cb.  meters. 


VOLUMES.      Cubic    and    Capacity   Measures   (concluded). 
Foreign. 

These  are  mostly  obsolete,  as  the  metric  system  is  now  used  in  most 
foreign  countries.  The  British  measures  are  included  among  the  U.  S. 
measures,  being  very  nearly,  and  sometimes  quite,  the  same.  The  trans- 
lated terms  are  merely  synonymous,  and  not  the  exact  equivalents. 

Germany.  Prussian.  1  Fuder  =  4  Oxhoft  of  1J^  Ohm  of  2  Eimer 
(bucket)  of  2  Anker  of  30  Quart  (Prussian).  1  Wispel  =  6  Tonne  (tun)  of 
4  Scheffel  of  16  Metzen  of  3  Quart  (Prussian).  1  Quart  (Prussian)  =  64 
cub.  Zoll  =  3^7  cub.  Fuss  =  1.14503  liters;  1  Wispel=  13.191  hectoliters; 

1  Tonne  =  2. 198  46  hectoliters;   1  Scheffel  =  0.549  61  hectoliter.     1  Schacht- 
ruthe=144.  cub.  Fuss  =  4.451  9  cub.  meters.      1  Klafter=108  cub.  Fuss  = 
3.338  9  cub.  meters. 

The  following  values  are  given  by  Nystrom;  they  are  in  cubic  inches. 
For  liquid  measures:  1  Stubgen  in  Bremen  =  194.5,  in  Hamburg  221,  in 
Hanover  231.  For  dry  measures:  1  Scheffel  in  Berlin  3  180,  in  Bremen 
4  339,  in  Hamburg  6  426.  1  Hanover  Matter  =  6  868. 

France.  "Old  Measures"  (systeme  ancien)  used  prior  to  1812.  1  muid 
(hogshead)  =  2  feuillettes  of  2  quartants  of  9  setiers  or  veltes  of  4  pots  of 

2  pintes  (pint,  though  more  nearly  equal  to  a  quart)  of   2  chopines  9f  2 
demi-setiers  of  2  possons  of  2  demi-possons  of  2  roquilles  (gill).     1  pinte 
<aucienne)  =  0.931  32  liter. 


VOLUMES.  55 

"New  measures"  (systeme  usuelle)  used  from  1812  to  1840.  1  boisseau  = 
2.751  2  gallons  (Brit.);  1  litron  (old  liter)  =  1.760  8  pints  (Brit.);  hence  it 
seems  that  1  boisseau  =  12^  litrons;  1  pinte  =  1  liter. 

The  following  values  are  given  by  Nystrom;  they  are  in  cubic  inches. 
For  liquid  measures:  1  Bordeaux  barrique=  14  033.  For  dry  measures: 
1  Marseilles  charge  =  9  411. 

Austria.  Liquid  measures:  1  Eimer  =  40  Maass  of  4  Seidel  of  2  Piff. 
1  Eimer  =  56.589  liters;  1  Maass- 1.414  724  liters  =  0.044  8  cub.  Fuss. 
Dry  measures:  1  Mut  or  Muth  =  30  Metzen  of  16  Maassel  of  4  Futter- 
maassel  of  2  Becher.  1  Metze  =  61.486  82  liters=  1.947  1  cub.  Fuss. 

The  following  values  are  given  by  Nystrom;  they  are  in  cubic  inches. 
For  liquid  measures:  1  Hungarian  Eimer  4474;  1  Trieste  Orne  =  4007; 
1  Vienna  Eimer  =  3  452;  1  Vienna  Maass  86.33.  For  dry  measures:  1 
Trieste  Stari  4  521;  1  Vienna  Metzen  3  753. 

Sweden.  1  am  =  4  anker  of  15  kannen  of  2  stop.  1  am  =  157.030  liters; 
1  kanne  =  2.617  liters=100  cub.  decimal  turn.  For  grain:  1  tonne  =  2 
spon  of  16  koppen.  1  tonne  =  56  kannen  =  146.565  liters.  According  to 
Nystrom  1  Swedish  eimer  =  4  794  cub.  inches;  1  kanna=159.57  cub. 
inches;  1  tunna  =  8  940  cub.  inches. 

Russia.  1  tschetwert  =  2  osmini  of  2  pajok  of  2  tschetwerik  of  4  tschet- 
werka  of  2  garnez.  1  tschetwert  =  209. 9  liters;  1  tschetwerik  =  26. 209 
liters;  1  tschetwerka  =  6.552  2  liters;  1  garnez  =  3. 276  1  liters.  1  botschka 
(barrel)  =  40  wedro  of  10  kruschky  or  stoof  of  10  tscharky.  1  kruschky  = 
1.228  5  liters.  According  to  Nystrom  1  Russian  weddras  =  752  cub.  inches; 

1  kunkas  =  94  cub.  inches;    1  Riga  loop  =  3  978  cub.  inches;    1  chetwert  = 
12  448.  cubic  inches. 

Spain.  1  cantara  of  wine  (castile)  =  4.263  gallons  (liquid;  U.S.);  in 
Havana  4.1  gallons.  1  fanega  of  corn  =  1.599  14  bushels  (U.  S.).  Accord- 
ing to  Nystrom  1  azumbras  =  22.5  cub.  inches;  1  quartellos  =  30.5  cub. 
inches;  1  catrize-=41  269.  cub.  inches;  1  Malaga  fanaga  =  3  783.  cub.  inches. 

Japan.  1  koku  =  10  to ;  1  to  =  10  sho  of  10  go  of  10  seki.  1  koku  =  180.39 
liters  =  47. 66  gallons  (U.  S.).  1  liter  =  0.005  54  koku.  1  gallon  (U.  S.)  = 
0.021  0  koku. 

Miscellaneous.  The  following  are  given  in  Nystrom's  Mechanics  in 
cubic  inches.  For  liquid  measures:  Amsterdam  Anker  =  2  331 ;  stoop  146; 
Copenhagen  Anker  2  355;  Antwerp  stoop  194.  Florence  oil  barille  1  946, 
wine  barille  2427;  Genoa  wine  barille  4530,  pinte  90.5;  Leghorn  oil 
barille  1  942;  Naples  wine  barille  2  544,  oil  stajo  1  133;  Rome  wine  barille 

2  560,  oil  barille  2  240,  boccali  80;  Sicily  oil  caffiri  662;  Malta  caffiri  1  270; 
Venice  Secchio  628.     Lisbon  almude  1  040 ;    Oporto  almude  1  555 ;    Con- 
stantinople   almude    319.     Geneva    setier    2  760.     Canaries    arrobas    949. 
Scotland  pint  =  103. 5.     Tripoli  mattari  1  376.     Tunis  oil  mattari  1  157. 

For  dry  measures:  Amsterdam  mudde  6596,  sack  4947;  Rotterdam 
sach  6361;  Copenhagen  toende  8489;  Antwerp  viertel  4705.  Florence 
stari  1  449;  Genoa  mina  7  382;  Leghorn  stajo  1  501,  sacco  4503;  Milan 
moggi  8444;  Naples  temoli  3122;  Rome  rubbio  16904,  quarti  4226; 
Sardinia  starelli  2988;  Sicily  salme  gros  21014,  salme  generate  16886; 
Malta  salme  16930;  Corsica  stajo  6014;  Venice  stajo  4945.  Lisbon 
alquiere  817,  fanega  3268;  Madeira  alquiere  684;  Oporto  alquiere  1  051. 
Alexandria  rebele  9  587,  kislos  10  418;  Constantinople  kislos  2  023;  Smyrna 
kislos  2141.  Algiers  tarrie  1219.  Tripoli  caffiri  19780;  Tunis  caffiri 
21  855.  Candia  charge  9  288.  Greece  medimni  2  390.  Persia  artaba 
4  013.  Poland  zorzec  3  120.  Geneva  coupes  4  739.  Scotland  firlot  2  197. 


56 


WEIGHTS    OR   MASSES. 


WEIGHTS  or  MASSES. 


The  fundamental  standard  of  mass  (commonly  called  weight)  of  the 
United  States  is  the  International  Kilogram,  a  cylindrical  mass  of  metal 
made  of  90%  platinum  and  10%  iridium,  and  preserved  at  the  Interna- 
tional Bureau  of  Weights  and  Measures,  near  Paris.  Copies  of  the  In- 
ternational Kilogram  are  possessed  by  each  of  the  20  countries  contrib- 
uting to  the  support  of  the  International  Bureau  of  Weights  and  Measures, 
and  these  copies  are  known  as  National  Prototypes.  The  United  States 
possesses  two  of  these  standards,  whose  values  in  terms  of  the  International 
Kilogram  are  known  with  the  greatest  accuracy.  One  of  the  kilograms 
known  as  No.  4  is  used  as  a  working  standard  and  the  other,  No.  20,  is 
kept  under  seal  and  only  used  to  check  No.  4.  One  of  the  objects  for  the 
maintenance  of  the  International  Bureau  of  Weights  and  Measures  is  to 
provide  for  the  recomparison  at  regular  intervals  of  the  various  National 
Prototypes  with  the  International  Kilogram,  thus  insuring  the  use  of  the 
same  standard  throughout  the  world. 

According  to  the  Act  of  Congress  of  July  28,  1866,  which  was  the  first 
general  legislation  upon  the  subject  of  fixing  the  standard  of  weights  and 
measures,  the  pound  is  to  be  derived  from  the  kilogram.  This  act  defined 
the  relation  1  kilogram  =  2. 204  6  avoirdupois  pounds;  but  this  has  since 
been  changed  to  the  more  accurate  value,  1  kilogram  =  15  432.356  39  grains, 
which  corresponds  to  2.204  622  34  avoirdupois  pounds,  or  1  avoirdupois 
pound  =  453. 592  427  7  grains.  This  value  is  the  one  now  used  by  the 
National  Bureau  of  Standards  in  Washington  and  the  avoirdupois  pounds, 
ounces,  grains,  etc.,  in  common  use  now  in  this  country  are  derived  from 
the  kilogram  according  to  this  relation,  and  are  consequently  fixed  and 
definite  units.  In  this  country  the  relation  between  the  pound  and  the 
kilogram  is  therefore  no  longer  to  be  determined  by  measurement ,  as  is 
often  supposed,  but  is  fixed  definitely  by  precise  definition. 

The  troy  j>ound  was  definitely  adopted  by  Congress  in  1828  for  coinage 
purposes  and  was  supposed  to  be  an  exact  copy  of  the  British  troy  pound; 
formerly  the  avoirdupois  pound  was  derived  from  this  according  to  the 
relation  1  avoirdupois  pound  700%7ao  troy  pound;  this  relation  is  still 
correct,  but  in  this  country  both  the  troy  and  the  avoirdupois  pounds  as 
now  standardized  by  the  Government  are  derived  from  the  kilogram  as 
stated  above,  and  hence  both  values  are  fixed  definitely.  The  old  troy 
pound,  "although  totally  unfit  for  such  purpose, "is  the  legal  standard  for 
coinage  purposes  in  this  country;  according  to  the  National  Bureau  of 
Standards,  the  troy  pound  of  the  Mint,  and  the  troy  pound  of  that  Bureau 
(based  on  the  kilogram)  are  the  same. 

The  kilogram  was  originally  intended  to  be  the  mass  of  a  cubic  deci- 
meter or  liter  of  pure  water  at  the  temperature  of  its  maximum  density. 
The  present  International  Prototype  Kilogram  is  an  exact  copy  of  the 
original  kilogram  of  the  archives,  which  when  made  was  supposed  to  be 
equal  to  the  weight  of  one  cubic  decimeter  of  water.  It  is  a  certain  mass 
of  platinum-iridium.  The  determination  of  the  precise  relation  of  this 
adopted  kilogram  to  the  mass  of  a  cubic  decimeter  of  water  is  now  in 
progress,  and  it  may  be  several  years  before  the  final  results  are  announced. 
The  results  thus  far  indicate  that  the  kilogram  is  heavier  than  it  would  be 
according  to  the  original  definition  by  about  25  milligrams,  or  about  25 
parts  in  1  000  000.  The  present  kilogram,  however,  is  definite,  and  the 
result  of  such  a  discrepancy  would  make  the  liter,  which  is  the  volume  of 
a  kilogram  of  water,  very  slightly  larger  than  the  cubic  decimeter  by  about 
25  parts  in  1  000  000.  The  question  of  correcting  the  kilogram  to  agree 
with  its  original  theoretical  definition  may  be  considered  when  the  deter- 
mination now  being  made  at  the  International  Bureau  has  been  com- 
pleted. For  all  but  the  most  refined  measurements,  however,  this  slight 
discrepancy  is  absolutely  negligible.  The  National  Bureau  of  Standards 
has  for  the  present  assumed  that  the  liter  and  the  cubic  decimeter  are 
equivalent;  this  identity  is  assumed  also  in  all  the  tables  in  this  book. 

The  relation,  legalized  in  Great  Britain  in  1898,  between  the  avoirdupois 
pound  and  the  kilogram,  is  precisely  the  same  as  that  in  this  country,  hence 
the  pounds  are  exactly  the  same  in  both  countries.  All  the  avoirdupois, 
troy  and  apothecary  weights  are  therefore  also  the  same  in  the  United 
States  and  in  Great  Britain. 


WEIGHTS    OR   MASSES.  57 

The  metric  weights  are  now  in  use  everywhere  for  all  accurate  scientific 
measurements.  In  this  country  they  are  coming  into  more  general  use; 
chemists  use  them  entirely.  The  avoirdupois  weights  (abbreviation  av.) 
are  used  for  most  purposes,  including  merchandise  in  general.  The  troy 
weights  are  used  for  weighing  gold,  silver,  etc.;  they  are  used  by  the  U.  S. 
Mint;  quantities,  even  much  larger  than  an  ounce,  are  usually  stated  in 
ounces  and  not  in  pounds.  In  the  apothecary  weights  (abbreviation  ap.) 
only  the  grain,  scruple,  and  dram  are  in  general  use  in  this  country;  the 
ounce  is  used  only  when  called  for  in  prescriptions;  apothecaries  almost 
always  use  the  avoirdupois  pound  and  ounce.  The  apothecary  ounce  and 
pound  are  the  same  as  the  troy.  The  grain  is  the  same  in  all  three  systems. 

No  fixed  rules  can  be  given  concerning  the  distinction  between  the  use 
of  the  short  or  net  ton  of  2  000  Ibs.  and  the  long  or  gross  ton  of  2  240 
Ibs.,  but  the  following  general  rules  may  serve  as  a  guide.  The  long 
ton  seems  to  be  the  only  official  one;  section  2951  of  the  Revised  Statutes 
of  the  U.  S.,  2d  Ed.,  1878,  Collection  of  Duties  upon  Imports,  Chapter  6. 
says  that  by  the  word  "ton"  in  that  chapter  is  meant  2  240  pounds.  With 
freight  on  railroads,  a  ton  of  2  240  pounds  seems  to  be  generally  used.  In 
the  iron  and  steel  trades,  pig  iron,  steel  rails,  and  iron  ore  are  bought  and 
sold  by  the  ton  of  2  240  pounds.  Coal  seems  to  be  weighed  in  long  tons 
also;  coke,  however,  is  weighed  in  short  tons  of  2  000  pounds.  The  short 
ton  of  2  000  pounds  seems  to  be  in  general  use  for  weighing  chemical  prod- 
ucts, or  in  general  for  the  more  expensive  products;  but  in  shipping  them 
on  railroads  the  ton  of  2  240  pounds  is  often  used.  The  short  ton  seems 
to  be  in  general  use  also  in  traction  engineering  calculations. 

What  are  popularly  termed  weights  should  more  correctly  be  called 
masses,  but  for  all  practical  purposes  the  two  terms  are  the  same.  The 
mass  of  a  body  is  the  same  anywhere  in  the  universe,  but  its  weight  depends 
on  the  attraction  of  gravitation.  The  mass  of  a  body  is  most  conveniently 
measured  by  its  weight,  and  if  the  attraction  of  gravitation  is  the  same 
(and  it  varies  but  slightly  at  different  parts  of  earth)  the  weight  will  be  a 
correct  measure  of  the  mass.  With  the  usual  beam  balance,  the  com- 
parison of  two  masses  by  means  of  their  weights  is  absolutely  exact  and 
quite  independent  of  the  value  of  gravity,  which  acts  equally  on  both; 
but  with  spring  scales  the  same  mass  may  have  slightly  different  weights, 
depending  on  the  force  of  gravity. 

When  weights  ate  considered  as  such,  and  not  as  masses,  they  are 
forces  and  have  the  dimensions  of  forces.  The  reduction  factors  in 
the  following  table  are  correct  whether  the  units  are  considered  as  masses 
or  as  forces.  The  reduction  factors  between  them  and  the  true  units  of 
force  (such  as  dyne  and  poundal)  are  given  in  a  separate  table  of  forces  so 
as  to  avoid  confusion.  See  also  note  under  Forces  and  under  Acceleration. 


WEIGHTS  or  MASSES.     (See  also  FORCES.)    Usual. 

**  Accepted  by  the  National  Bureau  of  Standards. 

*  Checked  by  L.  A.  Fischer,  Asst.  Phys.  National  Bureau  of  Standards. 

av.  means  avoirdupois.     Aprx.  means  within  2%. 

Logarithm 

1  milligram  [mg]  =  0.1    centigram 1-000  0000 

=  0.0154324*  grain.    Aprx.  %3-*- 10 2-1884322 

=  0.001    gram jj.QOO  0000 

1  centigram  [cg]=  10.  milligrams 1-000  0000 

=  0.154  324*  grain.    Aprx.  2/18 1.188  4322 

=  0.01    gram 2-000  0000 

1  grain  [gr]  =  same  in  avoirdupois,  troy,  or  apothecary  weights. 

=         64.798  9*  milligrams.    Aprx.  65 1-811  5678 

=         6.479  89*  centigrams.    Aprx.  6>£ 0-811  5678 

=   0.064  798  9*  gram.    Aprx.  1% -r- 100 2-811  5678 

=  0.002  285  71*  ounce  (av.).    Aprx.  »/4^- 1  000 3.359  0219 

1  decigram  [dg]=  1.543  24*  grains.    Aprx.  Ufa 0-188  4322 

=  0.1    gram LOOO  0000 


58  WEIGHTS    OR    MASSES. 

1  gram  [g]=  1  000.     milligrams 3-000  0000 

=  100.     centigrams 2-000  0000 

=  15.432  356  39**  grains.    Aprx.  151A 1-188  4322 

=      0.0352740*    ounce  (av.).    Aprx.  %-s- 100 2-5474541 

=     0.0321507*    ounce  (troy).    Aprx.  32 -«- 1  000 2-5071910 

=   0.002  204  62*    pound  (av.).    Aprx.  22-=-  10  000.  . .    3-343  3342 

=  0.001       kilogram 3-0000000 

1  ounce  [oz]  (av.)  =       2  834.95*  centigrams.    Aprx.  2A  X  10  000 .   3  452  5459 

=        437.500*  grains.    Aprx.  1%  X  100 2-640  9781 

=       28.349  5*  grams.    Aprx.  ft  X  100 1-452  5459 

=  16.  drams  (av.) 1-204  1200 

=     0.911  458*  ounce  (troy).    Aprx.  i°/n 1-959  7369 

=    0.062  500*  pound  (av.)  or  y80 2-7958800 

-0.028  349  5*  kilogram.    Aprx.  2A  -^  10 2-452  5459 

1  pound  [lb](av.)=  ?  000.**  grains 3-845  0980 

=  453.592  427  7  **  grams.    Aprx.  %  X  100.  .  .  .  2-656  6658 

=  256.       drams  (av.) 2-408  2400 

=  16.       ounces  (av.) 1-204  1200 

=  14,5833*    oz.  (troy).    Aprx.  Vr  X  100.    1-1638569 

=     0.4535924*    kilogram.    Aprx.  %-*•  10  ..   1-656  6658 
1  kilogram  or  kilo  [kg]: 

=  15432.35639**   grains.    Aprx.  3H  XI  000 4-1884322 

=  1  000.     grams 3-000  0000 

=  35.274  0*    ounces  (av.).    Aprx.  %  X  10 1-547  4541 

=  32.150  7*    ounces  (troy).    Aprx.  32 L507  1910 

=  2.204  62*    pounds  (av.).    Aprx.  2% 0-343  3342 

=          0.0220462*    hundredweight  (sh.).  Aprx.  22 -s- 1000.  2-343  3342 
=          0.0196841*    hundredweight  (long).    Aprx.  2  •*• 100. .   2-2941162 

=        0.001  102  31*    short  or  net  ton.    Aprx.  %  •*•  100 3-042  3042 

=  0.001      metric  ton 3-000  0000 

=      0.000  984  206      long  or  gross  ton.    Aprx.  1  -*•  1  000 4-993  0862 

1  hundredweight  [cwt]  (short): 

100.   pounds  (av.) 2-000  0000 

=       45.359  24*  kilograms.    Aprx.  %  X  10 ...   1-656  6658 

=       0.892  857*  hundredweight  (long).    Aprx.  %0 1-950  7820 

0.05    short  or  net  ton 2-698  9700 

=  0.045  359  24*  metric  ton.    Aprx.  ^2 2-6566658 

=   0.044  642  9*  long  or  gross  ton.    Aprx.  %  -*- 100 2-649  7520 

1  hundredweight  [cwt]  (long): 

112.  pounds  (av.).    Aprx.  1/9  X  1  000 2-049  2180 

=       50.802  4*  kilograms.    Aprx.  50 1-705  8838 

=       1.120  00*  hundredweights  (short).    Aprx.  1% 0-049  2180 

=     0.056  000*  short  or  net  ton.    Aprx.  %  -5- 10 2-748  1880 

=  0.050  802  4*  metric  ton.    Aprx.  Ho - 2-7058838 

0.05    long  or  gross  ton  or  Ho 2-6989700 

I  short  or  net  ton  [tn]: 

=         2  000.  pounds  (av.).  . . . 3-301  0300 

=     907.185*  kilograms.    Aprx.  900 2-957  6958 

=  20.  hundredweights  (short) 1-301  0300 

=    17.857  1*  hundredweights  (long).    Aprx.  %  X  10 1-251  8120 

=  0.907  185*  metric  ton.    Aprx.  subtract  Vio 1-957  6958 

=  0.892  857*  long  or  gross  ton.    Aprx.  subtract  %0 1-950  7820 

1  metric  ton,  tonne,  tonneau,  millier,  or  bar  [t]: 

=   2  204.62*  pounds  (av.).    Aprx.  22  X  100.  . 3-343  3342 

=         1  000.  kilograms 3-000  0000 

=   22.046  2*  hundredweights  (short).    Aprx.  22 1-343  3342 

=    19.684  1*  hundredweights  (long).    Aprx.  20 1-294  1162 

=    1.102  31*  short  or  net  tons.    Aprx.  add  Mo 0-0423042 

=  0.984  206*  long  or  gross  ton.    Aprx-.  1 1-993  0862 

1  long  or  gross  ton  [tn]: 

=       2240    pounds(av.).    Aprx.22XlOO. 3-3502480 

=  1  016.05*  kilograms.    Aprx.  1  000 3-006  9138 

=  22.400  0*  hundredweights  (short).    Aprx.  22 1-350  2480 

=  20.  hundredweights  (long) 1-301  0300 

=         1.12    short  or  net  tons.    Aprx.  1% 0-049  2180 

=  1.016  05*  metric  tons.    Aprx.  1 0-006  9138 


WEIGHTS    OR    MASSES. 


59 


Conversion  Tables  for  Weights. 

Note. — By  pounds  and  ounces  are  meant  avoirdupois  pounds  and  ounces. 


Grains  = 
Mil'grs  = 
Ounces  = 
Grams  = 
Pounds  = 
Klgms  = 
Sh.ton  = 

Lg.  ton  = 
Mt.ton  = 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

mlgrs 

grains 

gram 

ounces 

klgrms 

pound 

met. 
tons 

met. 
tons 

long 
tons 

0.984 
1.908 
2.953 
3.937 
4.921 
'5.905 
0.889 
7.874 
8.857 
9.842 

shrt 

tons 

04.799 
129.00 
194.40 
259.20 
323.99 
388.79 
453.59 
518.39 
583  19 
047.99 

0.015432 
0.030  805 
0.040  29  / 
0.001  730 
0.077  102 
0.092  594 
0.10803 
0.12340 
0.13889 
0.15432 

28.350 

)(>.(;:)',) 
S5.049 
113.40 
141.75 
170.10 
198.45 
220.80 
255.15 
283.50 

0.035  274 
0.070.518 
0.10582 
0.141  10 
0.17037 
0.21104 
0.24092 
0.282  19 
0.31747 
0.352  74 

0.453  59 
0.907  18 
1.3008 
1.8144 
2.2080 
2.721  0 
3.1751 
3.0287 
4.082  3 
4.5359 

2.204  0 
4.4092 
0.0139 
8.8185 
11.023 
13.228 
15.432 
17.037 
19.842 
22.040 

0.907 
1.814 
2.722 
3.029 
4.530 
5.443 
0.350 
7.257 
8.105 
9.072 

1.102 
2.205 
3.307 
4.409 
5.512 
0.014 
7.710 
8.818 
9.921 
11.02 

1.010 
2.032 
3.048 
4.004 
5.080 
0.090 
7.112 
8.128 
9.144 
10.10 

WEIGHTS  or  MASSES  (continued).     Unusual,  Special 
Trade,  or  Obsolete. 

av.  means  avoirdupois  weight;  ap.  means  apothecary  weight;  aprx. 
means  approximately. 

0.001  milligram  [r]  (has  no  name,  symbol  used  instead)  =0.000  015  432  4 
grains. 

1  jeweller's  grain  =  *4  carat  (diamond)  of  various  weights. 

1  carat  (diamond)  =  4.  jeweller's  grains  =  (according  to  Streeter)  in 
U.  S.  205.500  milligrams  or  3.171  4  grains,  and  in  England  205.409  milli- 
grams or  3.1700  grains;  other  authorities  give  3.168,  3.18,  and  3.2  grains 
in  U.  S.,  and  3.17  in  England.  For  values  in  other  countries  see  below 
under  Foreign  Weights. 

1  scruple  O]  (ap.)  =  20.  grains -1.295  978  (aprx.  %)  grams  =  M  dram 
(ap.)  =  ^4  ounce  (troy  or  ap.).  1  gram  =  0.771  618  (aprx.  %)  scruple. 

1  pennyweight  [dwt]  (troy)  =  24.  grains=  1.555  17  (aprx.  i^fr)  grams  = 
%o  ounce  (troy  or  ap.).  1  gram  =0.643  015  (aprx.  ^4i)  pennyweight. 

1  drachm,  same  as  dram. 

1  dram  (av.)  =  27%  or  27.34375  (aprx.  27^)  grains  =1.771  85  (aprx. 
%)  grams  =  0.455  729  (aprx.  ^li)  drams  (ap.)==1/i6  ounce  (av.).  1  gram  = 
0.564  383  'aprx.  #)  dram  (av.). 

1  dram  [  3  ]  (ap.)  =  60.  grains  =  3.887  934  grams  =  3.  scruples  =  2. 194  29 
(aprx.  !%)  drams  (av.)  =  Vs  ounce  (ap.).  1  gram  =0.257  206  drams  (ap.). 

1  decagram  or  dekagram  [dkg]=  154.323  6  grains  =  10.  grams  =  0.352  74 
ounce  (av.). 

1  ounce  (troy,  silk)  =  360.  grains  =  23. 327  6  grams. 

1  ounce  [oz]  (troy)  (used  chiefly  for  gold  and  silver)  =  480.  grains  = 
31.1035  grams  =  20.  pennyweights  =  1.097  14  (aprx.  H4o)  ounces  (av.)  =  l 
ounce  (ap.)=}42  or  0.0833333  pound  (troy  or  ap.)  =  0.068  571  4  pound 
(av.).  1  gram  =  0.032  150  7  ounce  (troy).  1  ounce  (av.)  =0.911  458  (aprx. 
10/ii)  ounce  (troy).  1  pound  (av.)  =  14.583  3  (aprx.  i°%)  ounces  (troy). 
1  kilogram  =  32. 150  7  ounces  (troy). 

1  ounce  [  §  ]  (ap.)  (used  only  in  prescriptions)  =480.  grains  =  31. 103  5 
grams  =  24.  scruples  =  8.  drams  (ap.)  =  1.097  14  (aprx.  1:Ho)  ounces  (av.)  = 
1  ounce  (troy)=Vi2  or  0.083  333  3  pound  (ap.  or  troy)  =0.068  571  4  pound 
(av.).  1  gram  =  0.032  1507  ounce  (ap.).  1  ounce  (av.)  =  0.911  458  (aprx. 
1(Hi)  ^unce  (ap.).  1  pound  (av.)  =  14.583  3  ounces  (ap.).  1  kilogram  = 
32.1507  ounces  (ap.). 


60  WEIGHTS   OR   MASSES. 

1  hectogram  [hg]  =  100.  grams  =  3.527  40  ounces  (av.). 

1  pound  (troy)  (seldom  used ;  troy  ounces  used  instead),  =  5  760.  grains  = 
240.  penny  weights  =  12.  ounces  (troy  or  ap.)  =  l  pound  (ap.)=576%ooo  or 
0.822  857  (aprx.  %)  pound  (av.)  =0.373  242  (aprx.  %)  kilogram.  1  pound 
(av.)  =  ™oo/5760  or  1.215  28  (aprx.%)  pounds  (troy).  1  kilogram  =  2. 679  23 
pounds  (troy). 

1  mint  pound  (U.  S.),  same  as  troy  pound. 

1  pound  (troy,  silk)  =  16.  ounces  (troy,  silk)  =  5  760.  grains  =  1  pound 
(troy). 

1  pound  (ap.)  (obsolete)  =  5  760.  grains  =  288.  scruples  =  96.  drams 
(ap.)  =  12.  ounces  (ap.  or  troy)  =  l  pound  (troy )=  570%ooo  orO. 822  857  (aprx. 
%)  pound  (av.)  =  0.373  242  (aprx.  %)  kilogram.  1  pound  (av.)=  ™°%7eo 
or  1.215  28  (aprx.  %)  pounds  (ap.).  1  kilogram  =2.679  23  pounds  (ap.). 

1  stone  (Brit.)  =  14.  pounds  (av.)  =  6.350  29  kilograms. 

1  myriagram  =  10  000.  grams  =  22.046  2  pounds  (av.)  =  10.  kilograms. 

1  quarter  [qr]  (short)  =  25.  pounds  (av.)  =  11.3398  kilograms  =  ^ 
hundredweight  (short). 

1  quarter  [qr]  (long)=28.  pounds  (av.)  =  12.700  6  kilograms  =  M  hun- 
dredweight (long). 

1  firkin  (butter)  =  56.  pounds  (av.)  (really  a  capacity  measure). 

1  bushel  (salt)  =  70.  pounds  (av.)  (really  a  capacity  measure). 

1  quintal  (av.)  =  100.  pounds  (av.)  =  45.359  24  kilograms  =1  hundred- 
weight (short). 

1  barrel  of  flour  (' 'legal"?)  =  196.  pounds  (av.). 

1  barrel  of  beef  or  pork  =  200.  pounds  (av.). 

1  quintal  (metric)  =  100.  kilograms  =  220.462  pounds  (av.). 

1  pig  (metal)  =  301.  pounds  (av.)  =  21H  stones. 

1  fother  (iron,  lead,  etc.)  =  2  408.  pounds  (av.)  =  172.  stones  =  8.  pigs. 

1  bloom  ton  =  lVio  long  tons  =  2  464.  Ibs. 

Relative  Weights  (used  in  chemistry). 

1  millimol  =0.001  mol  or  gram  molecule. 

1  mol  or  mole  =  1  gram  molecule,  which  see  below. 

1  gram  molecule  =  as  many  grams  of  a  substance  as  is  represented 
numerically  by  its  molecular  weight.  A  gram  molecule  of  any  gas  at  0°  C. 
and  760  mm  pressure  occupies  a  volume  of  22  380.  cubic  centimeters. 

1  kilogram  molecule  =  1  000.  gram  molecules. 

1  gramatom  =  as  many  grams  of  an  elemental  substance  as  is  repre- 
sented numerically  by  its  atomic  weight. 

WEIGHTS  or  MASSES  (concluded).     Foreign. 

These  are  mostly  obsolete,  as  the  metric  system  is  now  used  in  most 
foreign  countries.  The  British  measures  are  included  among  the  U.  S. 
measures,  being  very  nearly,  and  sometimes  quite,  the  same.  The  trans- 
lated terms  are  merely  synonymous,  and  not  the  exact  equivalents. 

Germany.  Prussia.  To  1839  inclusive:  1  Centner  (hundred weight )  = 
110  Pfund  (pound)  of  32  Loth  of  4  Quentchen.  1  Pfund  =  0.467  711  kilo- 
gram. From  1840  for  customs  and  from  1858  for  trade:  1  Centner  or 
Z.C.  =  100  Pfund  (pound)  of  30  Loth  of  10  Quentchen  of  10  Cent  or  zent  of 
10  Kern  or  Korn  (grain);  1  Pfund  =  0.5  kilogram.  Apothecaries  weight: 
1  Pfund  =  12  Unzen  (ounces)  of  8  Drachmen  of  3  Skrupel  of  20  Gran  (grain) 
(signs  the  same  as  in  U.  S.  apoth.  measure;  values  slightly  smaller); 
1  Pfund  =  0.350  783  kilogram  (see  also  under  Baden);  1  Schiffslast  (shipping 
weight)  =  40  Centner  =  2000  kilograms.  1  carat  (diamond)  in  Berlin 
205.440,  in  Frankfort  o.  M.  205.770,  and  in  Leipsic  205.000  milligrams 
\Streeter). 

Bavaria.  1  Pfund  (pound)  =  32  Loth  of  4  Quentchen;  1  Pfund  = 
0.560  0  kilogram. 

Saxony.  1  Pfund  (pound)  =  4  Pfenniggewicht  (pennyweight)  of  2 
Hellergewicht ;  1  Pfund  =  0.467  6  kilogram. 

Wurtemberg.  To  1850:  1  Pfund  (pound)  =  32  Loth  of  4  Quentchen  of 
4  Richtpfennig;  1  Pfund  =  0.467  7  kilogram.  Since  1850  like  in  Baden. 

Baden.  1  Pfund  (pound)  =  2  Mark  of  2  Vierlingen  of  4  Unzen  (ounces); 
Dr  1  Pfund  =  10  Zehnlingen  of  10  Centas  of  10  Dekas  of  10  As;  or  1  Pfund  = 


WEIGHTS    OR   MASSES.  61 

32  Loth  of  4  Quentchen;  1  Pfund  =  0.5  kilogram.  (Another  authority 
states  that  these  are  Prussian.) 

Hanover.     1  Pfund  =  0.4696  kilogram. 

The  following  additional  values  of  various  German,  pounds  in  pounds 
avoirdupois  are  given  by  Nystrom:  Berlin  1.033;  Bremen  1.100;  Bruns- 
wick 1.029;  Hamburg  1.068;  Hanover  1.073;  Leipsic  1.029. 

France.  Old  weights  (systeme  ancien  or  poid  de  marc)  used  prior  to 
1812.  1  millier  (ton)  =  10  quintaux  of  100  livres  (pounds);  1  livre  (pound) 
=  2  marcs  of  8  onces  (ounces)  of  8  gros  (drams)  of  3  deniers  (scruples)  of 
24  grains;  1  livre  =  0.489  506  kilogram.  "Usual"  weights  (systeme  usuel) 
used  from  1812  to  1840:  1  livre  (pound)  =  0.5  kilogram.  Apothecaries' 
weights:  1  livre  (romain)  (pound)  =  12  onces  (ounces)  of  8  dragmes  (drams) 
of  3  scrupules  (scruples)  of  20  grains;  1  livre  romain  =  0.75  livre  de  marc  = 
0.367  129  kilogram.  Nystrom  gives  1  Lyons  pound  (silk)  =  1.012  2  pounds 
av.  Streeter  gives  1  carat  (diamond)  =  205. 500  milligrams. 

Austria.  1  Centner  (hundred weight )  =  100  Pfund  (pound)  of  32  Loth 
of  4  Quentchen  of  4  Pfennig;  1  Pfund  =  0.560  01  or  0.56006  kilogram. 
1  Schiffstonne  (shipping  ton)  =  20  Centner=l  120.12  kilograms.  1  Meter- 
centner  (metric  hundredweight )  =  100  kilograms.  Nystrom  gives  1  Vienna 
pound  =  1.235  pounds  av.  Streeter  gives  1  carat  (diamond)  in  Vienna  = 
206.130  milligrams. 

Sweden.  1  centner  =  100  schalpfund  or  skalpund  or  mark  (pound)  of 
32  lod  of  4  kvintin  of  69^s  ass;  or  1  skalpund  =  100  korn  of  100  art;  1  skal- 
pund =100  korn  of  100  art;  1  skalpund  =  0.425  1  or  0.425  339  5  kilogram. 
1  schiffspund  (shipping  pound)  =  20  liespund  =  400  skalpund.  Nystrom 
gives  1  Swedish  pound  =  0.937  5  pound  av. ;  1  miner's  pound  =  0.828  6 
pound  av. 

Russia.  1  pfund  (pound)  =  32  loth  of  3  solotnick  of  96  doli;  lpfund  = 
0.409  512  or  0.409  531  kilogram.  Shipping  weights:  1  berkowitz  =  10  pud 
or  pood  of  40  pfund  (pounds);  1  berkowitz  =  163.81  kilograms.  Nystrom 
gives  1  Russian  pound  =  0.902  pounds  av. ;  1  Warsaw  pound  =  0.891  pound 
av. 

Switzerland.  Like  Baden  except  that  1  Pfund  =  32  Loth  of  16  Ungen. 
Nystrom  gives  1  Geneva  pound,  heavy,  =  1.214  pounds  av. 

Holland.  1  Amsterdam  or  Rotterdam  pound  =  1.089  pounds  av.  1 
carat  (diamond)  in  Amsterdam  =  205. 700  milligrams. 

Spain.  1  marco  or  mark  =  50  castellanos  of  8  tomines  of  12  Spanish 
gold  grains.  1  marco  =  0.507  6  pounds  av.  in  Spain  =  about  0.506  5 
pounds  av.  in  South  America;  other  values  up  to  0.54;  1  castellano  = 
71.07  to  71.04  grains  (U.  S.).  1  tonelada  (ton)  =  20  quintal  (hundred- 
weight) of  4  arroba  (quarters)  of  25  libra  (pounds);  1  tonelada  of  Castile  = 
2032.2  pounds  avoirdupois;  1  libra  =  0.461  0  kilogram  =  1.016  1  pounds 
av. ;  1  arroba  of  Castile  or  Madrid  =  25. 402  5  pounds  av. ;  it  has  various 
values  in  different  parts  of  Spain.  Nystrom  gives  1  Barcelona  pound  = 
0.888  1  pound  avoirdupois.  Streeter  gives  1  carat  (diamond)  =  205. 393 
milligram. 

Italy.  Nystrom  gives  the  following  values  in  pounds  avoirdupois: 
Bologna  pound  0.798;  Corsica  pound  0.759;  Florence  pound  0.749;  Genoa 
pound  1.077;  Leghorn  pound  0.749;  Naples  Rottoli  1.964;  Rome  pound 
0.748 ;  Sicily  pound  0.700 ;  Venice  pound,  heavy,  1 .055,  light,  0.667.  Streeter 
gives  1  carat  (diamond)  in  Florence  =  195. 200,  in  Leghorn  215.990  milli- 
grams. 

Japan.  1  kwan  =  1000  momme  of  10  fun.  1  kwan  =  6^  (aprx.)  kin 
of  160  momme.  1  kwan  =  3.76  kilograms.=  8.29  pounds  (av.).  1  kilo- 
gram =0.266  kwan.  1  pound  =  0.121  kwan. 

'  Miscellaneous.  Nystrom  gives  the  following  values  which  have  here 
been  reduced  to  pounds  avoirdupois:  Antwerp  pound  1.034;  Copenhagen 
pound  1.101;  Madeira  pound  0.698;  Tangiers  pound  1.061;  Cairo  rottoli 
0.952;  Alexandria  rottoli  0.935;  Algiers  rottoli  1.190;  Damascus  rottoli 
3.96;  Tunis  rottoli  1.110;  Tripoli  rottoli  1.120;  Cyprus  rottoli  5.24;  Can- 
dia  rottoli  1.164;  Aleppo  rottoli  4.89;  Aleppo  oke  2.79;  Constantinople 
oke  2.81;  Smyrna  oke  2.74;  Mocha  maund  3.00;  Morea  pound  1.101;  Ben- 

§al  seer  1.867;  Batavia  catty  1.302;  China  catty  1.326;  Japan  catty  1.300; 
umatra  catty  2.81.     Streeter  gives  1  carat  (diamond)  in  Lisbon  205.750, 
in  Borneo  105.000,  and  in  Madras  207.353  3  milligrams 


62  WEIGHTS    AND    LENGTHS. 

WEIGHTS  or  MASSES  and  LENGTHS;  WEIGHTS  of 
WIRES,  RAILS,  BARS;  FORCES  and  LENGTHS; 
FILM  or  SURFACE  TENSION;  CAPILLARITY. 

(Mass  -=-  length ;    force  ~-  length.) 

In  this  group  of  units  the  masses,  forces,  and  the  weights  considered  as 
^oth  masses  and  forces  have  all  been  combined  to  avoid  repetition  of  their 
Delations  to  each  other.  When  the  units  involve  masses  or  weights  con- 
>idered  as  masses,  as  in  pounds  per  foot,  they  are  used  to  measure  such 
quantities  as  the  weights  of  rails,  bars,  wires,  etc.;  while  when  the  units 
involve  forces  or  weights  considered  as  forces,  they  are  used  to  measure 
>uch  quantities  as  surface  tension,  capillarity,  etc.  The  dimensions  of  the 
units  in  the  two  cases  are  different.  Weights  considered  as  forces  involve 
the  value  of  gravity,  but  masses  and  forces  do  not. 
Aprx.  means  within  2%. 

Logarithm 
I  dyne  per  centimeter  [dyne/cm]: 

=      0.039  973  9  grain  per  inch.    Aprx. 4-^100 2-6017764 

=   0.001  019  79  gram  per  centimeter.    Aprx.  1  -5- 1  000 3-008  5098 

=  0.000  183  719  poundal  per  inch.    Aprx.  i%-5-  10  000 4-264  1530 

I  iTomid  (av.)  per  mile  [lb/ml]: 

=         0.281  849  kilogram  per  kilometer.    Aprx.  %• 1-450  0161 

=         0.110  480  grain  per  inch.    Aprx.  % 1-043  2829 

=   0.002  818  49  gram  per  centimeter.    Aprx.  %  -5- 100 3.450  0161 

=  0.000  568  182  pound  per  yard.    Aprx.  44-4-1  000 4-754  4873 

=  0.000  281  849  kilogram  per  meter.    Aprx.  2A  -*- 1  000 4-450  0161 

=  0.000  189  394  pound  per  foot.    Aprx.  19 -f- 100  000 4-277  3661 

1  kilogram   per  kilometer  [kg/km]  or  gram   per    meter 
[g/m]  or  milligram  per  millimeter  [mg/mm]: 

=  9.805  97  dynes  per  centimeter.    Aprx.  10. 0-991  4904 

=  3.548  00  pounds  per  mile.    Aprx.  % 0-549  9839 

=         0.391  983  grain  per  inch.    Aprx.  ^lo 1-593  2668 

0.01  gram  per  centimeter 2-000  0000 

=   0.002  015  91  pound  per  yard.    Aprx.  2-^1  000 3-304  4713 

=   0.001  801  54  poundal  per  inch.    Aprx.  %-*-!  000 3-255  6434 

0.001  kilogram  per  meter 3-000  0000 

=  0.000  671  970  pound  per  foot.    Aprx.  %•*-!  000 4-827  3500 

*    .rain  per  inch  [gr/in]: 

=  25.016  3  dynes  per  centimeter.    Aprx.  MX  100 1-398  2236 

=  9.051  4  pounds  per  mile.    Aprx.  9 0-956  7171 

=  2.551  14  kilograms  per  kilometer.    Aprx.  MX  10. .  .  .   0-406  7332 

=     0.025  511  4  gram  per  centimeter.    Aprx.  J£-5- 10 2-4067332 

=   0.005  142  86  pound  per  yard.    Aprx.  51  -5- 10  000 3-7112045 

=   0.004  595  96  poundal  per  inch.    Aprx.  6/is  -s- 100 3-662  3766 

=   0.002  551  14  kilogram  per  meter.    Aprx.  }<-*- 100 3-406  7332 

=     0.001  714  3  pound  per  foot.    Aprx.  17-s-lO  000 3-2340832 

'=•0.000  142  857  pound  per  inch.    Aprx.  3/<7  •*•  1  000 4-154  9020 

I  gram  per  centimeter  [g/cm]: 

=          980.597   dynes  per  centimeter.    Aprx.  1  000 2-991  4904 

=          354.800  pounds  per  mile.    Aprx.  %X100 2-549  9839 

100.  kg  per  km  or  g  per  m  or  mg  per  mm 2-000  0000 

=         39.198  3  grains  per  inch.    Aprx.  39 1-593  2668 

=       0.201  591   pound  per  yard.    Aprx.  % 1-304  4713 

=       0.180154  poundal  per  inch.    Aprx.  % -4- 10 1-2556434 

=  0.1   kilogram  per  meter 1-000  0000 

=   0.067  197  0  pound  per  foot.    Aprx.  %-*-10 2-827  3500 

=  0.005  599  75  pound  per  inch.    Aprx.  %  H- 100 3.748  1688 

1  pound  per  yard  [lb/yd]: 

1  760.  pounds  per  mile.    Aprx.  %  X  1  000 3-245  5127 

=        496.054  kilograms  per  kilometer.    Aprx.  MX  1  000..  .   2-6955287 

=        194.444  grains  per  inch.    Aprx.  %i  X  10  000 2-288  7955 

=       4.960  54  grams  per  centimeter.    Aprx.  5 0-695  5287 

=     0.496  054  kilogram  per  meter.    Aprx.  % 1-695  5287 

=     0.333  333  pound  per  foot  or  M 1-522  8787 

=  0.027  777  8  pound  per  inch.    Aprx.  i& -4-100 2-443  6975 


PRESSURES.  63 

1  poundalper  inch  =  5  443.11  dynes  per  cm.    Apr*.  !KX1  000  3-735  8470 
=  217.582  grains  per  inch.    Aprx.  HfcXIOO  2-337  6234 
=  5.550  81  grams  per  centimeter.  Aprx.  ^  0-744  356$ 
1  kilogram  per  meter  [kg/m]: 

=      3  548.00  pounds  per  mile.    Aprx.  %  X 1  000 3.549  983$ 

=  1  000.  kilograms  per  kilometer 3-000  OOOC 

=        391.983  grains  per  inch.    Aprx.  400 2-593  266$ 

=  10.  grams  per  centimeter 1-000  0000 

=      2.015  91  pounds  per  yard.    Aprx.  2 0-304  4713 

=    0.671  970  pound  per  foot.    Aprx.  ^ 1.827  3500 

=0.055  997  5  pound  per  inch.    Aprx.  ^-s-lOO 2-748  168ft 

1  pound  per  foot  [lb/ft]: 

=  5  280.  pounds  per  mile.    Aprx.  Vi9  X 100  000 3.722  6340 

=       1488.16  kilograms  per  kilometer.    Aprx.  %X  1  000.  ..   3-1726500 

=        583.333  grains  per  inch.    Aprx.  HT  X  10  000 2-7659168 

=       14.881  6  grams  per  centimeter.    Aprx.  %  X 10 1.172  6500 

=  3.  pounds  per  yard 0-477  1213 

=       1.48S  16  kilograms  per  meter.    Aprx.  % 0-172  6500 

=  0.083  333  3  pound  per  inch.    Aprx.  %  -*- 10 5-920  8188 

1  pound  per  inch  [lb/in]=       7  000.  grains  per  inch 3-845  0980 

=  178.579  gram/cm.    Aprx.  %X  100.  2-2518312 

=  36.  pounds  per  yard 1-556  3025 

=  17.8579  kgpermetr.    Aprx.  %X  10  1.251  8312 

12.  pounds  per  foot 1-079  1812 

Ton  per  mile.  A  term  popularly  though  incorrectly  used  for  "ton-mile, " 
which  see  under  units  of  Energy.  It  is  never  used  io 
the  sense  that  a  mile  of  something  weighs  a  ton,  except 
possibly  in  referring  to  submarine  cables,  rails,  etc. 


PRESSURES  j  PRESSURES  of  WATER,  MERCURY, 
and  ATMOSPHERE;  STRESS  or  FORCE  per  UNIT 
AREA.  WEIGHTS  or  FORCES  and  SURFACES; 
WEIGHTS  of  SHEETS,  DEPOSITS,  COATINGS, 

etc.     (Force -v- surface;  mass  ^-surface.) 

The  pressures  of  water  columns  have  all  (including  those  involving  only 
non-metric  units)  been  calculated  on  the  uniform  basis  that  a  cubic  deci- 
meter of  water  weighs  one  kilogram  (see  notes  under  Volumes  and  Weights). 

The  pressures  of  mercury  columns  have  all  been  calculated  on  the  basis 
that  the  specific  gravity  of  mercury  is  13,595  93,  which  is  the  value  accepted 
by  the  International  Bureau  of  Weights  and  Measures,  and  by  the  (U.  S.) 
National  Bureau  of  Standards.  This  is  the  value  used  in  the  legal  defini- 
tion of  the  liter  in  reference  to  the  atmospheric  pressure. 

The  pressure  of  the  atmosphere  here  used  as  a  standard  is  that  equal  to 
760  millimeters  of  mercury  of  the  specific  gravity  given  above. 

To  convert  barometric  pressures  from  millimeters  .to  inches  or  the  reverse, 
use  the  reduction  factors  for  one  millimeter  of  mercury  in  inches  of  mer- 
cury or  the  reverse. 

In  this  group  of  units,  the  forces,  masses,  and  the  weights,  considered  as 
both  forces  and  masses,  have  all  been  combined  to  avoid  repetition  of  their 
relations  to  each  other.  When  the  units  involve  forces,  or  weights  con- 
sidered as  forces,  they  are  used  to  measure  pressures  or  stresses  per  unit 
area;  while  when  the  units  involve  masses,  or  weights  considered  as 
masses,  they  are  used  to  measure  the  weights  of  sheets,  as  those  of  metals. 
For  instance,  pounds  per  square  foot  may  represent  a  pressure  or  the  weight 
of  a  sheet  of  metal;  in  the  former  case  the  pound  is  a  force  and  in  the 
latter  a  mass.  The  dimensions  of  the  units  in  the  two  cases  are  different. 
Weights  considered  as  forces  involve  the  value  of  gravity,  but  masses  and 
true  units  of  force  do  not. 

Aprx.  means  within  2%.     Hg  means  mercury. 


64 


PRESSURES. 


Logarithm 

Q.OOO  0000 
2-8273501 
2-008  5098 
2-000  0000 
3-319  8754 
5-875  1006 
4-668  9876 


1  dyne  per  square  centimeter  [dyne/cm2]: 

=  1.  barief i 

=     0.067  197  0  poundal  per  square  foot.    Aprx.  ^  -=- 10  .  .  . 

=     0.0101979  kilogram  per  square  meter.    Aprx.  ^oo .  .  •  • 

=  0.01   megadyne  per  square  meter 

=   0.002  088  70  pound  per  square  foot.    Aprx.  21  -4- 10  000. 

=  0.000  750  068  millimeter  of  mercury.    Aprx.  %  -r- 1  000. .  . 

=  0.000  466  646  poundal  per  square  inch.  Aprx.  iy3  •*•  10  000 
1  barie  f  (Fr.  barye)  ==  1.  dyne  per  square  centimeter. 
1  gram  per  square  decimeter  [g/dm2]: 

0.1  kilogram  per  sq.  m,  which  see  for  other  values. 

=  0.020  481  7  pound  per  square  foot.    Aprx.  205  •*• 10  000 .  .  . 
1  poundal  per  square  foot: 

=       14.881  6  dynes  per  square  centimeter.    Aprx.  %  X  10. . 

=     0.151  761  kilogram  per  square  meter.    Aprx.  %-r- 10..  .  . 

=0.031  083  2  pound  per  square  foot.    Aprx.  ^2 

=  0.011  162  2  millimeter  of  mercury.    Aprx.  %  +  10 

1  kilogram  per  square  meter  [kg/m2]: 

=  98.059  66  dynes  per  square  centimeter.    Aprx.  100. 

=  10.  grams  per  square  decimeter 

=  6.589  32  poundals  per  square  foot.    Aprx.  %  X  10. 

0.204  817  pound  per  sq.  foot.    Aprx.  205  -*- 1  000. .  . 

=  0.1  gram  per  square  centimeter 

=         0.073  551  4  millimeter  of  mercury.    Aprx.  %-r- 10  .  .  . 

=         0.045  759  2  poundal  per  square  inch.    Aprx.  %  -J- 100. 

=      0.003  280  83  foot  of  water.    Aprx.  Yz  -4- 100 

=       0.002  895  72  inch  of  mercury.    Aprx.  %  •*•  100 

=       0.001  422  34  pound  per  square  inch.    Aprx.  ^  •*• 100 .  . 

=  0.001  meter  of  water 

=    0.000  102  408  ton  (short )/sq.  ft.    Aprx.  102  -4-  1  000  000 

=  «     0.000  1   kilogram  per  square  centimeter 

=0.000  096  778  2  atmosphere.    Aprx.  29/30  -=_  1 00  000 

=  0.000  091  436  1   ton  (long)  per  sq.  ft.    Aprx.  ^4i  -=- 1  000 .  . 
1  megadyne  per  square  meter: 

=  100.  dynes  per  sq.  cm,  which  see  for  other  values 

1  pound  per  square  foot  [lb/ft2]: 

478.767  dynes  per  square  centimeter.    Aprx.  480. .  . 

=  48.824  1  grams  per  square  decimeter.    Aprx.  49 .... 

=  32.171  7  poundals  per  square  foot.    Aprx.  ^  X  1  000. 

=  4.882  41  kilograms  per  square  meter.    Aprx.  4% .  .  . 

=        0.359  108  millimeter  of  mercury.    Aprx.  ^ii 

=     0.016  018  4  foot  of  water.    Aprx.  %  -*- 100 

=     0.014  138  1  inch  of  mercury.    Aprx.  %  -r- 100 

=  0.006  944  44  pound  per  square  inch.    Aprx.  7-4-1  000 .  .  . 

=  0.004  882  41  meter  of  water.    Aprx.  49  •*•  10  000 

0.000  5  ton  (short)  per  square  fopt  or  ^  -5- 1  000 .... 

=  0.000  488  241  kilogram  per  sq.  cm.    Aprx.  49  -4- 100  000. .  . 

=  0.000  472  511  atmosphere.    Aprx.  47-4-100  000 

=  0.000  446  429  ton  (long)  per  sq.  foot.    Aprx.  %  -4-  1  000 .  .  . 
1  gram  per  square  centimeter  [g/cm2]: 

=  10.  kilograms  per  sq.  m,  which  see  for  other  values 1-000  0000 


2-000  0000 


t  Recommended  by  a  Committee  of  the  International  Physical  Congress 
of  1900,  in  Paris,  for  the  absolute  unit  of  pressure,  that  is,  for  one  dyne  per 
sq.  centimeter.  It  seems  it  was  not  officially  adopted  by  that  Congress. 
The  recommendation  includes  that  the  megabarie  (or  megabarye)  is  repre- 
sented with  sufficient  accuracy  for  practical  purposes  by  the  pressure  of 
75  cm  of  mercury  at  0°  C. ;  this  latter  is  nearly  what  is  usually  accepted  as 
the  pressure  of  one  atmosphere,  namely  76  cm  of  mercury.  It  was  origi- 
nally proposed  to  the  Congress  to  adopt  this  name  barie  for  the  atmos- 
pheric pressure,  making  it  equal  to  a  megadyne  per  sq.  centimeter,  but 
this  was  changed  by  the  Committee  of  that  Congress;  it  is,  however,  some- 
times used  in  this  sense. 


PRESSURES. 


65 


1  millimeter  of  mercury  column  [mm  Hg]: 

=         1  333.21    dynes  per  sq.  cm.    Aprx.  %  X  1  000  .......   3-124  8994 

=         89.587  9    poundals  per  square  foot.    Aprx.  90  .......   1-952  2495 

=    13.595  93f  kilograms  per  sq.  meter.    Aprx.  %  X  10  ----    1.133  4090 

—         2.784  68    pounds  per  square  foot.    Aprx.  li£  ........   0-444  7749 

=       0.622  138    poundal  per  square  inch.    Aprx.  %  ........   1-793  8870 

•  =   0.044  606  0    foot  of  water.    Aprx.  %-*•  10  .............   2-649  3932 

=   0.039  370  0    inch  of  mercury.    Aprx.  4-^100  ..........   2-595  1654 

=   0.019  3380    pound  per  square  inch.    Aprx.  19^-1  000.  .   2-286  4124 
=  0.013  595  93    meter  of  water.    Aprx.  %-:-100  ...........   2-133  4090 

=  0.001  392  34    ton  (short)  per  square  foot.  Aprx.  %  -*•  1  000.  3-143  7449 
=  0.001  359  59    kilogram  per  sq.  cm.    Aprx.  %  +  1  000  .....   3-133  4090 

=  0.001  315  79    atmosphere.     Aprx.Va^-l  000  ...........   3-119  1864 

=  0.00124316    ton  (long)  per  sq.  foot.    Aprx.  ^-s-  100  ----   3-0945269 

1  poundal  per  square  inch: 

-       2  142.95  dynes  per  sq.  cm.    Aprx.  %4  X  10  000  ........ 

=       21.853  6  kilograms  per  square  meter.    Aprx.  1%'XlO.  - 
=       1.607  36  millimeters  of  mercury.    Aprx.  %  ........... 

=  0.031  083  2  pound  per  square  inch.    Aprx.  3/32  .......... 

1  foot  of  water  column  =        304.801  kgpersq.  m.  Aprx.  300.. 
-       62.428  3  Ibs/sq.  ft.  Aprx.  5/^xlOO 
22.4185mm  Hg.  Aprx.%X  10... 


.  .  . 

0.882  617  inch  Hg.     Ap.  subtr.  1/9. 
0.433  530  Ib  per  sq.  inch.  Aprx. 


3-331  0124 
1-339  5220 
0-206  1130 
2-492  5253 
2-484  0158 
1.795  3817 
1.350  6068 
1.945  7722 
.  .          .  .     -    1-637  0192 

0.304  801  meter  of  water.  Aprx.3/io  1-484  0158 
=  0.031  214  2  sh.  ton/ft2.  Aprx.  M2.  -  -   2-494  3517 
"  =0.030  480  1  kg/cm2.  Aprx.  3-^100.  .   2-484  0158 

=  0.0294980  atm.    Aprx.  3  -MOO..  .  .    2-4697932 

=  0.027  869  8  1.  ton/ft2.  Ap.  lli-t-WQ..  2-445  1337 
1  inch  of  mercury  column  [in  Hg]: 

=        345.337  kilograms  per  sq.  meter.    Aprx.  %  X  100  .....   2-538  2436 

=       70.731  0  pounds  per  square  foot.    Aprx.  70  ..........    L849  6095 

=     25.400  05  millimeters  of  mercury.    Aprx.  MX  100  ......   L404  8346 

=       1.132  99  feet  of  water.    Aprx.  1%  ...................   0-054  2278 

=     0.491  187  pound  per  square  inch.    Aprx.  }/>,  ...........    1-691  2470 

=     0.345  337  meter  of  water.    Aprx.  %0  .................    1-538  2436 

-0.035  365  5  ton  (short)  per  sq.  foot.    Aprx.  %-t-  100  ......   2-548  5795 

=  0.034  533  7  kiiogram  per  sq.  centimeter.    Aprx.  %  •*•  100.  .   2-538  2436 
=  0.033  421  1  atmosphere.    Aprx.  Mo  ....................   2-524  0210 

=  0.031  576  3  ton  (long)  per  sq.  foot.    Aprx.  V^  ...........   2-499  3615 

1  pound  per  square  inch  [lb/in2]: 

=        703.067  kilograms  per  sq.  meter.    Aprx.  700  ........   2-846  9966 

=  144.  pounds  per  square  foot.    Aprx.  ^  X  1  000  ----   2-158  3625 

=       51.711  6  millimeters  of  mercury.    Aprx.  3Ve  X  10  .....    1-713  5876 

=       32.171  7  poundals  per  sq.  inch.    Aprx.  Mi  X  1  000  ----    1.507  4746 

=       2.306  65  feet  of  water.    Aprx.  %  ...................   0-362  9808 

=       2.035  88  inches  of  mercury.    Aprx.  2  ...............   0-308  7530 

=     0.703  067  meter  of  water.    Aprx.  7/lo  ................   1-846  9966 

0.072  ton  (short)  per  square  foot.    Aprx.  ^4  ......    2-857  3325 

=  0.0703067  kilogram  per  sq.  centimeter.    Aprx.  7  -5-  100.  .  2-8469966 
=  0.068  041  5  atmosphere.    Aprx.  Mo  ...................   2-832  7740 

:  =0.0642857  ton  (long)  per  sq.  foot.    Aprx.%1^-10  .......   2-8081145 

1  meter  of  water  column  or 

)  metric  ton  per  square  meter  [t/m2]: 

=          1  OOO.  kilograms  per  square  meter  ............  ....   3-000  0000 

=        204.817  pounds  per  square  foot.    Aprx.  205  .........   2-3113659 

=       73.551  4  millimeters  of  mercury.    Aprx.  MXlOO  .....    1.866  5910 

=       3.280  83  feet  of  water.    Aprx.  MX  10  ...............   0-515  9842 

=       2.895  72  inches  of  mercury.    Aprx.  2%  .............   0-461  7564 

=       1.422  34  pounds  per  sq.  inch.    Aprx.  #  X  10  .........   0-1530034 

=     0.102  408  ton  (short)  per  sq.  foot.    Aprx.  4H+  100  .....   1-010  3359 

0.1   kilogram  per  square  centimeter  ............   1-000  0000 

-0.096  778  2   atmosphere.    Aprx.  97-^1  000  .............   2-985  7774 

=  0.091  436  1   ton  (long)  per  square  foot.     Aprx.  Hi  ......   2-061  1179 


t  This  is  the  specific  gravity  of  mercury  used  throughout  in  these  tables. 


66  PRESSURES. 

1  ton  (short)  per  square  foot  [tn/ft2]: 

=          9  764.82  kilograms  per  square  meter.    Aprx.  9  800.  .    3-989  6641 

=  2  000.  pounds  per  square  foot 3-301  0300 

=          718.216  millimeters  of  mercury.    Aprx.  %  X  I'OOO  .  .    2-856  2551 

=         32.036  7  feet  of  water.    Aprx.  32 1.505  6483 

=         28.276  2  inches  of  mercury.     Aprx.  %  X  100 1-451  4205 

=         13.888  9  pounds  per  square  inch.     Aprx.  %  X  10.  ... 

=         9.764  82  meters  of  water.    Aprx.  98 -^  10 

=       0.976  482  kilogram  per  sq.  cm.    Aprx.  subtract  Mo  .  •  • 

•  =       0.945  021   atmosphere.    Aprx.  subtract  ^o 

=       0.892  857  ton  (long)  per  sq.  ft.    Aprx.  subtract  % 

=  0.006  944  44  ton  (short)  per  sq.  inch.    Aprx.  7  +  1  000...  . 
=  0.006  200  40  ton  (long)  per  sq.  inch.     Aprx.  %  -5- 100 

1  kilogram  per  square  centimeter  [kg/cm2]: 

=       10  000.  kilograms  per  square  meter 4-000  0000 

=   2  048.17  pounds  per  square  foot.    Aprx.  2  050 3.311  3659 

=     735.514  millimeters  of  mercury.    Aprx.  MX  1  000 2-866  5910 

=   32.808  3  feet  of  water.    Aprx.  MX  100 1.515  9842 

=   28.957  2  inches  of  mercury.    Aprx.  ty  X  100 1-461  7564 

=    14.223  4  pounds  per  sq.  inch.    Aprx.  Vr  X  100 1-153  0034 

=  10.  meters  of  water 1-000  0000 

=    1.02408  tons  (short)  per  sq.  foot.    Aprx.  add  ^o 0-0103359 

=  0.967  782  atmosphere.     Aprx.  subtract  Mo 1-985  7774 

=  0.914  361   ton  (long)  per  square  foot.    Aprx.  i%i 1-961  1179 

=         0.001   metric  ton  per  square  centimeter 3-000  0000 

1  barie  f  =  75  centimeters  of  Hg  (aprx.).    Accurately  75.0068. 
"          =1  megadyne  per  square  centimeter. 

1  me^aburiet  =  1  megadyne  per  square  centimeter 0-000  0000 

1  megadyne  per  sq.  cm.  =750.068  mm.  of  Hg.   Aprx.  ^XlOOO  2-8751006 
—0.986 931  atmosphere  (stand.)   Ap.  1  1-9942870 

1  atmosphere  [atm]  (standard): 

=    10  332.9  kilograms  per  square  meter.    Aprx.  10  300 4-014  2226 

=   2  116.35  pounds  per  square  foot.    Aprx.  2  100 3-325  5885 

76O.  millimeters  of  mercury.    Aprx.  MX  1  000 2-880  8136 

=   33.900  6  feet  of  water.    Aprx.  }  JX  100 1-530  2068 

=   29.921  2  inches  of  mercury.     Aprx.  bJ 1-475  9790 

=    14.696  9  pounds  per  square  inch.    Aprx.  *% 1-167  2260 

=    10.332  9  meters  of  water.    Aprx.  10M 1-014  2226 

=    1.058  18  t9ns  (short)  per  sq.  foot.    Aprx.  add  Mo 0-024  5585 

=    1.033  29  kilograms  per  sq.  cm.    Aprx.  add  Mo 0-014  2226 

=    1.013  24  megadynes  per  sq.  centimeter.    Aprx.l 0-005  7130 

=    1.013  24  megabaries.J     Aprx.  1 0-005  7130 

=  0.944801   ton  (long)  per  sq.  foot.    Aprx.  subtract  Mo- ...   1-9753405 

1  ton  (long)  per  square  foot  [tn/ft2]: 

=          10  936.6  kilograms  per  sq.  meter.    Aprx.  1 1  000 4.038  8821 

2  240.  pounds  per  square  foot.    Aprx.  %  X  1  000  .  .    3-350  2480 

=          804.402  millimeters  of  mercury.    Aprx.  800 2-905  4731 

=         35.8811   feet  of  water.    Aprx.  %i  X  100 1-5548663 

=         31.669  3  inches  of  mercury.    Aprx.  32 1-500  6385 

=         15.555  6  pounds  per  square  inch.    Aprx.  3M 1-191  8855 

=         10.936  6  meters  of  water.    Aprx.  11 1.038  8821 

=  1.12  tons  (short)  per  sq.  foot.    Aprx.  add  1A  •  -  -  .   0-0492180 

=         1.093  66  kilograms  per  square  cm.    Aprx.  add  Vio  .  .  .   0-038  8821 

=         1.058  42  atmospheres.    Aprx.  add  Mo 0-024  6595 

=  0.007  777  78  ton  (short)  per  sq.  inch.    Aprx.  %  - 100 §-890  8555 

=  0.006  944  44  ton  (long)  per  sq.  inch.    Aprx.  7-M  000  .  .  .   3-841  6375 

1  kilogram  per  square  millimeter  [kg/mm2]: 

=  100.  kilograms  per  sq.  cm,  which  see  for  other  values.  .  .  .   2-000  0000 

1  ton  (short)  per  square  inch  [tn/in2]: 

144.  tons  (short)  per  sq.  foot.    Aprx.  ty  X  1  000 .  . 

=     140.613  kilograms  per  sq.  cm.    Aprx.  %  X  100 

=     136.083  atmospheres.    Aprx.  %  X  100 

=  0.892  857  ton  (long)  per  sq.  inch.    Aprx.  subtr.  Mo  ... 
=  0.140  613  metric  ton  per  sq.  centimeter.    Aprx.  ty 

t  Not  authoritative;    see  foot-note  on  page  64. 
t  Authoritative;  see  foot-note  on  page  64. 


WEIGHTS   AND   VOLUMES. 


67 


1  ton  (long)  per  square  incli  [tn/in2): 

=     157.487  kilograms  per  sq.  centimeter.    Aprx.  i#  X  100  .  2-197  2446 

=     152.413   atmospheres.    Aprx.  %  X  100 2-183  0220 

144.  tons  (long)  per  sq.  foot.    Aprx.  V7  X  1  000 2-158  3625 

1.12  tons  (short)  per  square  inch.    Aprx.  add  Y%  .  .  .  0-049  2180 

=  0.157  487  metric  ton  per  sq.  cm.    Aprx.  1$*10 1-197  2446 

1  metric  ton  per  square  centimeter  [t/cm2]: 

=      1  000.  kilograms  per  sq.  cm,  which  see  for  other  values..  3-000  0000 

-967.782  atmospheres.    Aprx.  970 2-985  7774 

=  7.111  70  tons  (short)  per  sq.  inch.     Aprx.5/rX10 0-8519734 

=  6.349  73  tons  (long)  per  sq.  inch.    Aprx.  %  X  10 0-802  7554 

Conversion   Tables   for  Pressures. 


Pounds  per 

sq.  inch  = 

kilogram 

atmos- 

Kilograms 

persq.cm. 

Ibs.  per 

pheres. 

per  sq.  cm  = 

sq.  in. 

atmos- 

Atmosph's = 

Ibs.  per 

kg.  per 

pheres. 

sq.  in. 

sq.  cm. 

1 

0.070  307 

14.223 

14.697 

0.068042 

1.0333 

0.967  78 

2 

0.14061 

28.447 

29.394 

0.13608 

2.066  6 

1.9356 

3 

0.21092 

42.670 

44.091 

0.204  12 

3.099  9 

2.903  3 

4 

0.281  23 

56.894 

58.788 

0.272  17 

4.1332 

3.871  1 

5 

0.351  53 

71.117 

73.485 

0.34021 

5.1665 

4.8389 

6 

0.421  84 

85.340 

88.181 

0.408  25 

6.1997 

5.8067 

7 

0.492  15 

99.564 

102.88 

0.47629 

7.2330 

6.774  5 

8 

0.562  45 

113.79 

117.58 

0.544  33 

8.266  3 

7.742  3 

9 

0.632  76 

128.01 

132.27 

0.61237 

9.2996 

8.7100 

10 

0.70307 

142.23 

146.97 

0.680  42 

10.333 

9.6778 

WEIGHTS  or  MASSES  and  VOLUMES  ;  DENSITIES; 
WEIGHTS  of  MATERIALS;  MASSES  per  unit  of 
VOLUME.  (Weight  ~  volume.) 

Only  the  more  usual  units  are  given  here,  as  the  table  would  otherwise 
have  become  very  long  and  cumbersome.  The  relations  between  such 
compound  units  as  these  are  the  same  as  those  between  their  individual 
units  whenever  one  of  the  latter  is  the  same  in  both;  for  instance,  the  rela- 
tion between  pounds  per  cubic  yard  and  kilograms  per  cubic  yard  is  the  same 
as  between  pounds  and  kilograms,  and  as  these  are  given  in  the  tables  of 
weights  they  are  not  repeated  here.  In  such  a  reduction  multiply  the 
pounds  per  cubic  yard  by  the  value  of  1  pound  in  kilograms.  Similarly, 
the  relation  between  pounds  per  cubic  yard  and  pounds  per  cubic  meter  is 
the  same  as  that  between  a  cubic  meter  and  a  cubic  yard,  but  in  this  case 
care  must  be  taken  in  the  reduction  on  account  of  the  word  "  per,"  not  to 
multiply  the  former  by  the  value  of  one  cubic  yard  in  cubic  meters,  but  to 
divide  instead,  as  a  cubic  yard  is  smaller  than  a  cubic  meter,  hence  *he 
weight  per  cubic  meter  is  larger.  To  avoid  such  a  long  division  use  instead 
the  reciprocal  relation,  namely,  the  value  of  one  cubic  meter  in  cubic  yards 
and  then  multiply.  The  general  rule  for  all  compound  units  is  that  if  the 
individual  unit  to  be  changed  is  preceded  by  the  word  "  per,"  then  divide 
by  the  value  of  the  old  unit  in  terms  of  the  new  one  (or  multiply  by  its  re- 
ciprocal); in  all  other  cases  multiply,  even  when  the  unit  follows  a  hyphen, 
as,  for  instance,  in  the  case  of  pounds  in  foot-pounds. 

In  this  group  of  units  the  weights  are  always  masses  and  never  forces; 
no  unit  exists  having  the  dimensions  of  force  divided  by  volume.  The 
value  of  gravity  is  therefore  not  involved  in  these  values. 

"Weights  of  Materials.  In  the  metric  system  the  number  represent- 
ing the  density  or  specific  gravity  also  represents  the  actual  weight  in  grams 
of  a  cubic  centimeter  of  the  material.  Hence  the  actual  weight  of  any  other 


68  WEIGHTS   AND    VOLUMES. 

unit  of  volume  of  that  material  in  terms  of  any  other  unit  of  weight  is  deter- 
mined by  merely  multiplying  the  specific  gravity  or  density  (when  based 
on  water,  as  is  usual  for  all  materials  except  gases)  by  the  value  of  1  gram 
per  cubic  centimeter  in  terms  of  those  units  as  given  in  the  table  below. 
Thus  the  weight  of  any  material  in  pounds  per  cubic  foot  is  62.43  multiplied 
by  its  specific  gravity,  this  figure  62.43  being  the  value  of  1  gram  per  cubic 
centimeter  in  terms  of  pounds  per  cubic  foot.  Similarly,  the  specific 
gravity  or  density  is  easily  calculated  by  means  of  the  figures  in  this  table, 
when  the  weight  of  a  unit  of  volume  is  given  in  terms  of  any  of  the  usual 
non-metric  units.  Thus  if  the  weight  of  any  material  is,  say,  100  pounds 
per  cubic  foot,  its  specific  gravity  is  0.016  02  (which  is  the  value  of  1  pound 
per  cubic  foot  in  terms  of  grams  per  cubic  centimeter  in  the  table)  multi- 
plied by  100,  that  is  1.602. 
Aprx.  means  within  2%. 

Logarithm 
1  pound  per  cubic  yard  [lb/yd3]: 

=         0.593  273  kilogram  per  cubic  meter.    Aprx.  %o  ......   1-773  2545 

=     0.037  037  0  or  HT  Pound  per  cb.  ft.    Aprx.  %-*  10 2-5686362 

=  0.000  593  273  gram  per  cb.  cm  or  kg  per  lit.    Ap,  .0006. .  .   4-773  2545 

=  0.000  593  273  ton  (met.)  per  cb.  meter.  Aprx.  6-*-.  10  000.   4.773  2545 

0.000  5  ton  (short)  per  cubic  yard  or  ^-^1  000.  .  .  .    4-698  9700 

=  0.000  446  429  ton  (long)  per  cubic  yard.    Aprx.  %  +  l  000.  4-649  7520 

1  kilogram  per  cubic  meter  [kg/m3]: 

1.685  56  pounds  per  cubic  yard.    Aprx.  *% 0-226  7455 

=     0.062  428  3  pound  per  cubic  foot.    Aprx.  ^-f-10.  .....   2-795  3817 

0.001  gram  per  cb.  cm  or  kilogram  per  liter 3-000  0000 

=-=  0-001  ton  (met.)  per  cubic  meter 3-000  0000 

=  0.000  842  782  ton  (short)  per  cb.  yd.    Aprx.  %-*- 1  000. .  .   4-925  7155 

=  0.000  752  484  ton  (long)  per  cb.  yd.    Aprx.  %•*-  1  000 4-876  4975 

1  grain  per  cubic  inch  [gr/in3]: 

=       0.246  857  pound  per  cubic  foot.    Aprx.  }i 1-392  4457 

=  0.003  954  25  gram  per  cb.  cm  or  kg  per  lit.    Aprx.  "H  ooo...  3-597  0640 
1  pound  per  bushel  [Ib/bu]  (U.  S.): 

=   12.871  8  kilograms  per  cubic  meter.    Aprx.  9AX10 1-1096387 

=    1.287  18  kilograms  per  hectoliter.    Aprx.  % 0-1096387 

=    1.032  02  pounds  per  bushel  (Brit.).    Aprx.  add  Ko- 0-013  6888 

=  0.803  564  pound  per  cubic  foot.    Aprx.  % 1-905  0204 

1  pound  per  bushel  [Ib/bu]  (Brit.): 

=    12.472  4  kilograms  per  cubic  meter.    Aprx.  YsX  100.  . .  .   1-095  9499 

=    1.247  24  kilograms  per  hectoliter.    Aprx.  1% 0-095  9499 

=  0.968  972  pound  per  bushel  (U.S.).    Aprx.  subtr.  Mo L986  3112 

=  0.778  630  pound  per  cubic  foot.    Aprx.  % 1-891  3318 

1  kilogram  per  hectoliter  [kg/hi]: 

=  10.  kilograms  per  cubic  meter,  which  see  for  other  values.    1-000  0000 
1  pound  per  cubic  foot  [lb/ft3]: 

=  27.  pounds  per  cubic  yard.    Aprx.  %  X  10 1-431  3638 

=       16.018  4  kilograms  per  cubic  meter.    Aprx.  %X  10.  ..    1-204  6183 

=       4.050  93  grains  per  cubic  inch.    Aprx.  4 0-607  5543 

=       1.601  84  kilograms  per  hectoliter.    Aprx.  % 0-2046183 

=       1.284  31   pounds  per  bushel  (Brit.).    Aprx.  % 0-108  6684 

=       1.244  46  pounds  per  bushel  (U.  S.).    Aprx.  1M 0-094  9796 

=    0.160  538  pound  per  gallon  (Brit.).    Aprx.  %-hlO 1-205  5784 

=  0.133681  pound  per  gallon  (liquid;  U.S.).  Aprx.%-5- 10  Ll26  0683 
=  0.0160184  gram  per  cb.  cm  or  kg  per  lit.  Aprx.  % -MOO..  2-2046183 
=  0.0160184  ton  (met.)  per  cubic  meter.  Aprx.  %-^  100. .  2-2046183 
=  0.013  5  ton  (short)  per  cubic  yard.  Aprx.  4/3 -H  100 ..  2-1303338 
=  0.012  053  6  ton  (long)  per  cubic  yard.  Aprx.  %-r-lOO  .  .  2-081  1158 
1  pound  per  gallon  [Ib/gal]  (liquid;  U.  S.): 

=   7.480  52  pounds  per  cubic  foot.    Aprx.  MX10.. 0-8739317 

=    1.200  91  pounds  per  gallon  (Brit.).    Aprx.  add  % 0-0795101 

=  0.119  826  gram  per  cb.  cm  or  kg  per  lit.    Aprx.  12 -i- 100...  1.078  5500 
1  pound  per  gallon  [Ib/gal]  (Brit.): 

=       6.229  05  pounds  per  cubic  foot.    Aprx.  634 0-794  4216 

=  0.832  702  4  pound  per  gallon  (liquid;  U.  S.).    Aprx.  %..  .    1.920  4899 

=  0.099  779  2  gram  per  cb.  cm  or  kg  per  liter.    Aprx.  Ho...  .   2-9990399 

1  pound  per  quart  [lb/qt]=4.  pounds  per  gallon 0-602  0600 


WEIGHTS   AND    VOLUMES    OF   WATER.  69 

1  gram  per  cubic   centimeter  [g/cm3]  or 

1  kilogram  per  liter  [kg/1]  or 

1  ton  (met.)  per  cubic  meter  [t/m3]: 

=       1685.57  pounds  per  cubic  yard.    Aprx.  K  X 10  000. . .   3-2267455 

1  000.  kilograms  per  cubic  meter ,  .  .  .  .    3-000  0000 

=        252.893   grains  per  cubic  inch.    Aprx.  MX  1  000 2-402  9360 

100.  kilograms  per  hectoliter 2-000  0000 

=       80.177  1   pounds  per  bushel  (Brit.).    Aprx.  80 1-904  0501 

=       77.6893  pounds  per  bushel  (U.  S.).    Aprx.  %X100..   1-8903613 

=       62.428  3  pounds  per  cubic  foot.    Aprx.  ^>(  100 1-795  3817 

=       10.022  1   pounds  per  gallon  (Brit.).    Aprx.  10 1-000  9601 

=       8.34545  pounds  per  gal  (liquid;  U.S.).  Aprx. Vi2  X  100  0-921  4500 
=    0.842  783  ton  (short)  per  cubic  yard.    Aprx.  subtr.  K  •   1-925  7155 

=    0.752  484  ton  (long)  per  cubic  yard.    Aprx.  »£ 1-876  4975 

=  0.0361275  pound  per  cubic  inch.     Aprx.  "Hi  -f- 10 2-5578380 

0.001   kilogram  per  cubic  centimeter 3-000  0000 

1  ton  (short)  per  cubic  yard  [tn/yd3]: 

=    1  186.55  kilograms  per  cubic  meter.    Aprx.  %  X  1  000. .  .   3-074  2845 

=     118.655  kilograms  per  hectoliter.    Aprx.  %X  100 2-074  2845 

=   95.133  7  pounds  per  bushel  (Brit.).    Aprx.  95 1-978  3346 

=   92.181  9  pounds  per  bushel  ( U.  S.).    Aprx.  Vn  X  1  000. .  .    1-964  6458 

=   74.074  1  pounds  per  cubic  foot.    Aprx.  MX  100 1-869  6662 

=    1.186  55  tons  (met.)  per  cb.  m  or  kg  per  lit.    Ap.  add  %...  0-074  2845 
=  0.892  857  ton  (long)  per  cubic  yard.    Aprx.  9/i0 1-950  7820 

1  ton  (long)  per  cubic  yard  [tn/yd3]: 

=  1  328.93  kilograms  per  cubic  meter.    Aprx.  %  X  1  000 3-123  5025 

=  132.893  kilograms  per  hectoliter.    Aprx.  %X  100 2-123  5025 

=  106.550  pounds  per  bushel  (Brit.).    Aprx.  107 2-027  5526 

=  103.244  pounds  per  bushel  (U.  S.).    Aprx.  103 2-013  8638 

=  82.963  0  pounds  per  cubic  foot.    Aprx.  %  X 100. , 1-918  8842 

=  1.328  93  tons  (met.)  per  cb.  m  or  kg  per  lit.    Ap.  add  K--  0-123  5025 
=         1.12  tons  (short)  per  cubic  yard.    Aprx.  add  % 0-049  2180 

1  pound  per  cubic  inch  [lb/in3]: 

=       27.679  7  grams  per  cb.  centimeter.    Aprx.  ^X  10.  ...    1-442  1620 
==0.027  679  7  kilogram  per  cubic  centimeter.  Aprx.  ^-^  100  2-442  1620 

1  kilogram  per  cubic  centimeter  [kg/cm3]: 

=       1  000.  grams  per  cb.  cm  or  tons  (met.)  per  cb.  m 3-000  0000 

=  36.127  5  pounds  per  cubic  inch.    Aprx.  36 1-557  8380 

WEIGHTS  and  VOLUMES  of  WATER. 

Factors  for  calculating  •weights  or  volumes  of  materials  from 

their  specific  gravity. 

The  following  two  groups  of  numbers  give  the  weights  (W)  of  all  the 
different  units  of  volume  of  water  occurring  in  practice;  also  the  volumes 
(V)  of  all  the  different  units  of  weight  of  water  occurring  in  practice;  the 
latter  are,  of  course,  the  reciprocals  of  the  former  and  are  given  so  as  to 
avoid  the  long  divisions  by  the  former. 

Besides  their  direct  application  to  hydraulics  and  to  the  calibration  of 
vessels  and  for  measuring,  or  for  the  indirect  determinations  of  irregular 
volumes  by  means  of  weights,  they  are  also  of  use  for  determining  th-3 
weights  of  materials,  as  the  weight  of  a  unit  of  volume  of  any  material, 
whether  solid  or  liquid,  is  its  specific  gravity  or  density  multiplied  by  one 
of  these  factors,  W;  or  the  volume  of  a  unit  of  weight  of  any  material, 
whether  solid  or  liquid,  is  one  of  these  factors,  V,  divided  by  its  specific 
gravity  or  density.  They  are  applicable  also  to  gases  provided  the  value 
of  the  specific  gravity  or  density  which  is  used  is  based  on  water  and  not 
on  air  or  hydrogen. 

For  the  weights  of  columns  of  water,  mercury,  or  the  air,  see  under 
Pressures. 

All  these  values,  even  those  given  entirely  in  English  units,  have  been 
calculated  from  the  uniform  bases  that  one  liter  of  water  weighs  one  kilo- 
gram, and  that  a  liter  is  equal  to  a  cubic  decimeter.  (See  notes  on  the  liter 
in  the  introductory  remarks  on  units  of  Volume  and  Weight.) 


70  WEIGHTS   AND    VOLUMES    OF    WATER. 

WEIGHTS  of  WATER,  W. 

Aprx.  means  within  2%. 

Logarithm 

1  cubic  centimeter  =*         15.423  4  grains.    Aprx.  33^or  151^.  .  1.188  4322 

=                      1.  gram 0-000  0000 

=   0.0352740  oz  (av.).    Aprx.  %-?- 100  ..  2-547454] 

=  0.00220462  Ib  (av.,).    Aprx.  %  + 100.  ..  3.348  3342 

I  cubic  inch  =        252.893  grains.    Aprx.  MX1  000 2-402  9360 

=       16.387  2  grams.    Aprx.  %  X  100  or  »% 1.214  5038 

=  0  .578  040  ounce  (av.).    Aprx.  ^ 1.761  9579 

-=0.036  127  5  pound  (av.).    Aprx.  #10 2-557  8380 

1  pint  (liquid;  U.  S.)  =    1.043  18  pounds  (av.).    Aprx.  add  Ho.  .  0-018  3600 

=0.473  179  kilogram.    Aprx.  Ki  X  10 1-675  0258 

1  pint  (dry;  U.  S.)  =    1.213  90  pounds  (av.).    Aprx.  % 0-0841813 

=  0.550  614  kilogram.    Aprx.  ^^-10 1-7408471 

1  pint  (Brit.)  =         1.252  77  pounds  (av.).    Aprx.  % 0-097  8701 

=  0.568  245  39  kilogram.    Aprx.  ty? 1-754  5359 

1  quart  (liquid;  U.  S.)  =   2.086  36  Ib  (av.).    Aprx.  2VW  or2y10.  .  Q-319  3900 

=  0.946  359  kilogram.    Aprx.  subt.  Ho-  .  .  1-9760558 

1  liter  =  2.204  62   pounds  (av.).    Aprx.  s^0 0-343  3342 

1.  kilogram 0-000  0000 

1  quart  (dry;  U.S-)  =  2.427  79  pounds  (av.).    Aprx.  2^0  or  1%..  0-3852113 

=  1.101  23  kilograms.    Aprx.  add  Ho 0-041  8771 

1  quart  (Brit.)=       2.505  53  pounds  (av.).    Aprx.  1% 0-398  9001 

=  1.136  490  8  kilograms.    Aprx.  add  Vr 0-055  5659 

1  gallon  (liquid;  U.  S.)  =8.345  45  pounds  (av.).    Aprx.  5% Q-921  4500 

=  3.785  43  kilograms.    Aprx.^XlO 0-5781158 

1  gallon  (Brit.)  =       10.022  1  pounds  (av.).    Aprx.  10 1-0009601 

=  4.545  963  1  kilograms.    Aprx.  %  or  A1A 0-657  6259 

1  peck  (U.  S.)  =  19.422  3  pounds  (av.).    Aprx.  %i  XI  000 1-2883013 

=  8.809  82  kilograms.    Aprx.J^XlO 0-9449671 

1  peck  (Brit.)=       20.044  3  pounds  (av.).    Aprx.  20 1-301  9901 

=  9.091  926  2  kilograms.    Aprx.  9  or  .i°<Hi 0.958  6559 

1  cubic  foot  =       62.428  3  pounds  (av.).    Aprx.  ^XlOO 1-7953817 

=       28.317  0  kilograms.    Aprx.  %  X  100 1-452  0475 

=0.031  214  2  ton  (short).    Aprx.  ^2 2-494  3517 

=  0.028  317  0  ton  (met.).    Aprx.  2Ao 2-452  0475 

=  0.027  869  8  ton  (long).    Aprx.  ^oo -.  -  2-445  1337 

1  bushel  (U.  S.)  =  •  77.689  3  pounds  (av.).    Aprx.  %  X  100 1-890  3613 

=  35.239  28  kilograms.    Aprx.  35 1-547  0271 

1  bushel  (Brit.)=         80.177  1  pounds  (av.).    Aprx.  80 1-904  0501 

=  36.367  704  8  kilograms.    Aprx.Vn  X  100 L560  7159 

1  hectoliter  =        220.462  pounds  (av.).    Aprx.  220 2-343  3342 

=                100.  kilograms 2-000  0000 

=     0.110  231  ton  (short).     Aprx.  % 1-042  3042 

=                 0.1   ton  (met.) 1-000  0000 

=  0.098  420  6  ton  (long).    Aprx.  Vio 2-993  0862 

1  cubic1  yard  =    1  685.56  pounds  (av.).    Aprx.  M  X  10  000 3-226  7455 

=     764.559  kilograms.    Aprx.  %X  1  000 2-883  4113 

=0.842  782  ton  (short).    Aprx.  % 1-925  7155 

=  0.764  559  ton  (met.).    Aprx.  % |-883  4113 

=  0.752  484  ton  (long).    Aprx.  M 1-876  4975 

1  cubic  meter  =   2  204.62  pounds  (av.).    Aprx.  2  200 3-343  3342 

=         1  000.  kilograms 3-000  0000 

=    1.102  31   tons  (short).    Aprx.  add  Ho 0-042  3042 

=                1.  ton  (met.) 0-000  0000 

••              =0.984  206  ton  (long).    Aprx.  1 1-993  0862 


WEIGHTS    AND    VOLUMES    OF   WATER. 


71 


VOLUMES    of  WATER,    V, 

1  grain  =   0.064  798  9  cubic  centimeter.    Aprx.  !%  -s- 100 2-811  5878 

=  0.003  954  25  cubic  inch.     Aprx.4-^-1000 3-597  0640 

1  gram  =  1.  cubic  centimeter 0-000  0000 

=  0.061  0234  cubic  inch.    Aprx.6-^-100 5-7854962 

1  ounce  (av.)  =  28.349  5  cubic  centimeters.    Aprx.  2/r  X  100 1-452  5459 

=  1.729  98  cubic  inches.    Aprx.  % 0-238  0421 

1  pound  (av.)  =  453.592  cb.  centimeters.    Aprx.  %  X  100 .  .   2-656  6658 

=  27.679  7  cubic  inches.    Aprx.  ^X  10 1-4421620 

=-        0.958  606  pt (liquid;  U.S.).  Aprx.  subtr.  l/2Q-  1-981  6400 

=         0.823  794  pint  (dry;  U.S.).    Aprx.% 1-9158187 

=         0.798  233  pint  (Brit.).    Aprx.%  or8/io 1-902  1299 

=         0.479  303  qt  (liquid;  U.  S.).  Aprx.  10-^21..      .680  6100 

.     =         0.453  592  liter.    Aprx.  10-^22  or  e/n ".656  6658 

=         0.411  897  quart  (dry;  U.  S.).     Aprx.  i%4.  .   ".614  7887 

=         0.399  117  quart  (Brit.).    Aprx.  4/40 601  0999 

=         0.119  826  gal  (liquid;  U.S.).  Aprx.  12 -r- 100.  ".Q78  5500 

=     0.099  779  2  gallon  (Brit.).    Aprx.  Vio 999  0399 

=     0.051  487  1  peck  (U.  S.).    Aprx.  51 -^-1  000..  .   2-711  6987 

=     0.049  889  6  peck  (Brit.).    Aprx.  lAo 2-6980099 

=     0.016  018  4  cubic  foot.    Aprx.  %-=- 100 2-2046183 

=     0.012  871  8  bushel  (U.  S.).    Aprx.  %-*- 100.  ..   2-109  6387 

=     0.012  472  4  bushel  (Brit.).    Aprx.  %Q 2-095  9499 

=   0.004  535  92  hectoliter.    Aprx.  »/2-r-l  000 3.656  6658 

=  0.000  593  273  cubic  yard.    Aprx.  6  •*- 10  000 4-773  2545 

=  0.000  453  592  cubic  meter.    Aprx.  %-s- 10  000.  .   4-656  6658 

1  kilogram  =  1  000.  cubic  centimeters 3-000  0000 

=         61.023  4  cubic  inches.    Aprx.  60 1-785  4962 

=         2.11336  pints  (liquid;  U.  S.).    Aprx.  sy10  ...   Q-324  9742 

=         1.816  15  pints  (dry;  U.  S.).    Aprx.  *%i Q-259  1529 

=         1.759  80  pints  (Brit.).    Aprx.  % 0-245  4641 

=         1.05668  quarts  (liquid;  U.S.).  Aprx.  add  Ho-   0.023  9442 

=  1.  liter 0-000  0000 

=       0.908  078  quart  (dry;  U.  S.).    Aprx.  9/io 1-958  1229 

=       0.879  902  quart  (Brit.).    Aprx.  % 1-944  4341 

=       0.264  170  gallon  (liquid ;  U.S.).    Aprx.  8/30 1-421  8842 

=       0.219  975  gallon  (Brit.).    Aprx.  22 H- 100 1-342  3741 

=      0.113510  peck  (U.S.).    Aprx.  %  •*•  10 1-0550329 

=      0.109988  peck  (Brit.).    Aprx.  11 -4- 100 1-0413441 

=  0.035  314  5  cubic  foot.    Aprx.  %•-*- 100 2-547  9525 

=   0.0283774  bushel  (U.  S.).    Aprx.  %-t-10 2-4529729 

=  0.027  496  9  bushel  (Brit.).    Aprx.  1^-5-100 2-439  2841 

=  0.01   hectoliter 2-000  0000 

=  0.00130794  cubic  yard.    Aprx.  %  •*•  1  000 3-1165887 

==  0.001   cubic  meter 3-000  0000 

i  ton  (short)  =    32.036  7  cubic  feet.    Aprx.  32 1-505  648C 

-   9.071  85  hectoliters.    Aprx.  9 0-957  6958 

*=    1.186  55  cubic  yards.    Aprx.% 0-0742845 

,     =0.907  185  cubic  meter.    Aprx.  9Ao 1-957  6958 

1  ton  (  netric)  =  35.314  5  cubic  feet.    Aprx.  %X10 1.547  9525 

=  10.  hectoliters 1-000  0000 

=  1.307  94  cubic  yards.    Aprx.  % 0-116  5887 

=  1.  cubic  meter 0-000  0000 

1  ton  (long)  =  35.881  1  cubic  feet.    Aprx.  %i  X  100 1.554  8663 

=  10.160  5  hectoliters.    Aprx.  10 1-006  9138 

=  1.328  93  cubic  yards.    Aprx.  % „ 0-123  5025 

'*  — 1.016  05  cubic  meters.    Aprx.  1.  . .  t 0-006  9138 


72  ENERGY;  WORK;  HEAT. 


ENERGY;  WORK;  HEAT;  VIS-VIVA ;  TORQUE. 

(Force X length;  mass X temperature;  elec.  quant. X e.  m.  f.) 

All  energy  units  or  measures  are  here  grouped  together,  be  they  mechan- 
ical, electrical,  thermal,  chemical,  kinetic,  potential,  etc.  Units  of  power, 
not  being  energy  but  rate  of  energy,  are  not  included  in  this  group  (see  note 
under  units  of  Power).  For  the  relations  between  energies  stated  in  terms 
of  power  units,  as  horse-power-hours,  and  kilowatt-hours,  see  the  relations 
between  horse-powers  and  kilowatts  under  Power. 

The  relations  between  the  mechanical  and  thermal  units  of  energy  are 
based  on  the  mechanical  equivalent  of  heat;  those  between  the  mechanical 
and  electric  or  absolute  units  are  based  on  the  value  of  gravity;  those  be- 
tween the  thermal  and  the  electrical  or  absolute  units  are  based  on  the 
specific  heat  of  water  in  absolute  units.  These  three  bases  are  the  three 
connecting  links  between  these  three  classes  of  energy  units.  The  three 
links  are  again  interlinked  by  the  relation  that  the  mechanical  equivalent 
of  heat  multiplied  by  gravity  is  equal  to  the  specific  heat  of  water,  when  all 
three  are  reduced  to  grams,  centimeters,  and  seconds.  This  will  be  found 
explained  more  fully  under  Inter-relations  of  Units  in  the  Introduction. 

Torque  is  a  force  multiplied  by  a  lever-arm;  it  is  a  "moment"  and  is 
therefore  measured  in  units  like  foot-pounds  or  kilograms-meters;  the 
relation  between  unitr;  of  torque  are  therefore  the  same  as  those  in  the 
table  between  the  same  named  units  of  energy.  In  order  to  show  whether 
torque  or  energy  is  meant  by  such  similarly  named  units,  it  has  become 
customary  to  reverse  the  accepted  term  whenever  torque  is  meant;  that  is, 
to  use  the  term  pound-feet  for  torque  and  foot-pounds  for  energy ;  also  meter- 
kilograms  for  torque  and  kilogram-meters  for  energy;  this  distinction  is  to 
be  recommended  even  though  the  length  factor  unfortunately  comes  first 
in  foot-pounds  of  energy  and  last  in  kilogram-meters  of  energy.  For  the 
relations  between  units  of  torque  and  units  of  energy,  see  the  end  of  the 
following  table.  Torque  multiplied  by  an  angle  is  energy,  and  as  an  angle 
has  no  dimensions,  it  follows  that  the  dimensions  of  torque  and  of  energy 
are  the  same,  notwithstanding  the  fact  that  there  is  no  direct  equivalent 
between  a  foot-pound  of  torque,  for  instance,  and  a  calorie  of  energy.  If 
the  torque  is  measured  statically  it  is  performing  no  work,  yet  it  is  a  true 
torque;  it  is  then  like  a  suspended  weight  which  is  capable  of  doing  work 
when  released;  but  this  weight  (force)  must  be  multiplied  by  a  length  or 
distance  through  which  it  acts  in  order  to  give  energy,  while  torque  already 
includes  this  factor  length;  this  apparent  discrepancy  arises  from  the  fact 
that  an  angle,  by  which  torque  must  be  multiplied  to  reduce  it  to  energy, 
has  no  dimensions.  In  units  of  energy  the  length  factor  is  in  the  direction 
of  the  force,  while  in  units  of  torque  it  is  perpendicular  to  the  force.  The 
two  lengths  are,  therefore,  at  relatively  different  angles  with  each  other, 
which  explains  how  the  angle  enters  into  the  difference  in  the  nature  of  the 
two  units;  torque  differs  from  energy  somewhat  like  the  so-called  "wattless 
component"  differs  from  the  true  energy  in  electrical  quantities.  See  also 
the  relations  given  at  the  end  of  the  following  table. 

Mechanical  Equivalent  of  Heat,  and  the  Heat  Unit.  The  me- 
chanical equivalent  of  heat,  sometimes  called  Joule's  equivalent,  is  an 
empirical  number  which  gives  the  relation  that  heat  units  bear  to  mechani- 
cal units,  thus  enabling  one  to  calculate  how  much  mechanical  energy  stated 
in  foot-pounds,  kilogram-meters,  horse-power-hours,  etc.,  is  equal  to  any 
given  amount  of  heat  energy  stated  in  calories  or  thermal  units,  or  the 
reverse. 

The  mechanical  units  are  all  based  on  the  C.  G.  S.  or  absolute  system  of 
units  and  on  the  acceleration  of  gravity,  while  the  heat  units  are  based  on 
an  inherent  property  of  water,  and  on  the  thermometer  scale  and  kind  of 
thermometer.  The  two  bases  are  therefore  entirely  different,  and  the  re- 
lation between  them,  namely,  the  mechanical  equivalent  of  heat,  therefore 
is,  and  must  always  remain,  one  that  has  to  be  determined  by  experiment. 
Moreover,  these  experiments  are  intricate  and  the  heat  units  themselves 
arc  not  yet  definitely  and  accurately  established,  owing  to  the  variations 
H  the  specific  heats  of  water  between  0°  and  100°  C.  and  to  the  differences 
in  the  kinds  of  thermometers;  therefore  the  mechanical  equivalent  of  heat 


E 


ENERGY;   WORK;  HEAT.  73 

is  not  yet  known  to  very  great  accuracy,  although  the  accuracy  is  quite 
sufficient  for  all  purposes  except  perhaps  for  very  refined  physical  research. 
Quite  a  numper  of  researches  have  been  made,  among  which  are  a  few  very 
accurate  ones,  but  the  most  authoritative  value  is  unquestionably  the  one 
recommended  by  Griffiths  and  by  Ames  in  their  reports  to  the  International 
Physical  Congress  of  1900,  which  met  in  Paris  (Rapports,  Congrcs  Inter- 
national de  Physique,  1900,  tome  1,  pp.  226  and  204).  Griffiths  there 
concludes,  after  a  careful  comparison  and  discussion  of  the  best  determi- 
nations, to  recommend  the  number  4.187  joules  for  the  calorific  capacity 
(more  generally  called  the  specific  heat)  of  1  gram  of  water  raised  from  15° 
to  16°  C.,  measured  on  the  hydrogen  scale  of  the  International  Bureau. 
The  probable  error,  he  says,  is  le.ss  than  1  in  2000.  He  also  recommends 
that  this  be  considered  the  same  as  the  mean  value  per  degree  between  0° 
and  100°  C.,  and  believes  it  to  be  very  improbable  that  the  error  in  this 
assumption  attains  2  in  1  000.  This  therefore  also  defines  the  unit  of  heat 
which  he  recommends  as  the  intermediate  thermic  standard.  The  value 
which  he  recommends,  4.187,  agrees  with  that  given  as  the  most  probable 
in  the  report  of  Ames,  after  reduction  to  the  same  temperature  interval. 
In  Ames'  report  all  the  important  determinations  of  the  mechanical  equiva- 
lent of  heat  are  discussed  and  compared.  This  value  is  the  mean  of  the 
determinations  of  Rowland,  Griffiths,  Schuster,  and  Gannon,  and  Callender 
and  Barnes,  when  all  their  results  are  reduced  to  the  hydrogen  scale  of 
temperature  and  when  the  electrical  methods  are  corrected  for  the  proba- 
ble error  of  the  present  international  volt  as  now  legally  fixed  in  terms  of 
the  Clark  cell. 

Taking  for  the  value  of  gravity  at  sea-level  and  at  45°  latitude  9.805  966 
in  meters  (Helmert,  Die  math.  u.  phys.  Theorien  der  hoehern  Geodaesie, 
II,  p.  241,  1884),  this  specific  heat  reduces  to  426.985  kilogram-meters 
as  the  value  of  the  mechanical  equivalent  of  heat.  According  to  Griffiths' 
probable  error,  the  true  value  therefore  lies  between  427.20  and  426.77 
kilogram-meters,  showing  that  the  fourth  figure  is  still  uncertain.  Some 
recent,  very  carefully  made  researches  by  Barnes,  which  were  not  finished 
in  time  to  be  included  in  Griffiths'  report,  give  the  value  426.6,  and  it  is 
probable,  therefore,  that  the  true  value  is  lower  than  Griffiths'  probable 
mean,  rather  than  higher,  and  may  be  even  nearer  to  426.6  than  to  426.985. 
But  as  the  final  value  in  Griffiths'  report  may  be  considered  as  semi-official, 
the  author  has  adopted  that  value  throughout  this  book,  except  that  in 
order  to  make  it  approach  rather  than  depart  from  Barnes'  value,  it  has 
been  abbreviated  to  426.9  instead  of  427.0.  In  view  of  the  fact  that  the 
fourth  place  is  still  uncertain  it  would  not  be  rational  to  retain  more  than 
four  places  of  figures. 

This  naturally  also  establishes  the  absolute  value  of  the  heat  units  and 
of  the  mean  specific  heat  of  water  to  be  used  in  this  book.  The  value  of  the 
specific  heat  of  water  in  absolute  units  is  the  same  thing  as  the  mechanical 
equivalent  of  heat  stated  in  ergs  or  joules.  The  specific  heat  of  water  is 
different  at  different  temperatures  between  0°  and  100°  C.,  but  the  value 
between  15°  and  16°  C.  (according  to  Barnes  at  16°  C.)  is  as  nearly  as  has 
been  determined  equal  to  the  mean  per  degree  for  the  whole  value  between 
0°  and  100°  C.,  and  a  heat  unit  based  on  this  meah  is  therefore  much  more 
definitely  defined  than  if  based  merely  on  a  rise  of  temperature  of  one  degree 
without  stating  which  degree. 

Although  it  would  perhaps  be  more  rational  to  consider  the  specific  heat 
of  water  to  be  the  fundamental  quantity,  the  mechanical  equivalent  of  heat 
being  then  derived  from  it,  yet  the  author  has  reversed  this  by  adopting  a 
more  simple  abbreviated  number  for  the  mechanical  equivalent,  and  letting 
the  specific  heat  be  the  incommensurable,  derived  quantity.  This  was 
done  because  the  former  is  used  very  frequently,  while  the  latter  is  not. 
The  uncertainty  of  the  fourth  place  of  figures  in  either  of  these  values, 
which  may  be  ±2,  does  not  warrant  any  fine  distinction  between  the  funda- 
mental and  the  derived  value.  The  resulting  value  of  the  specific  heat  of 
water  then  becomes  4.186  17  instead  of  4.187.  In  view  of  the  researches 
of  Barnes  above  mentioned,  the  former  value  is  probably  even  more  nearly 
correct  than  the  latter. 

The  value  of  the  mechanical  equivalent  of  heat  and  that  of  the  calorie 
Or  heat  unit  adopted  in  this  book,  are  therefore  given  by  the  following 
statements:  426.9  kilogram-meters  of  energy  will  raise  1  kilogram  of  water 
from  15°  to  16°  C.,  hydrogen  scale,  a*  sea-level,  latitude  45°,  and  are  there' 


74  ENERGY;  WORK;  HEAT. 

fore  equivalent  to  one  large  calorie  or  kilogram  calorie.  The  small  calorie 
or  gram  calorie,  equal  to  one  thousandth  of  the  large  calorie,  is  the  amount 
of  heat  that  will  raise  the  temperature  of  1  gram  of  water  from  15°  to  16°  C., 
hydrogen  scale;  this  is  taken  as  equal  to  one  hundredth  of  the  amount  of 
heat  that  will  raise  the  temperature  of  1  gram  of  water  from  0°  to  100°  C. 
Unfortunately  writers  generally  do  not  state  which  of  the  two  calories  they 
mean.  The  thermal  unit,  or  British  thermal  unit,  or  BTU,  is  the  amount 
of  heat  which  will  raise  one  pound  (av.)  of  water  one  degree  Fahrenheit;  in 
these  tables  it  is  a  derived  unit  whose  value  is  determined  from  that  of 
either  of  the  calories.  The  hybrid  unit  based  on  the  pound  and  the  Centi- 
grade scale  has  no  name. 

This  mechanical  equivalent,  namely  426.9  kilogram-meters  per  kilogram 
Centigrade  heat  unit,  corresponds  to  778.104  foot-pounds  per  pound  Fah- 
renheit heat  unit  or  per  thermal  unit.  The  probable  error  in  them  is 
thought  to  be  within  1  in  2  000.  For  further  equivalents  or  converions 
factors  based  on  this,  see  the  table. 


ENERGY;  WORK;  HEAT;  VIS-VIVA;  TORQUE. 

Aprx.  means  within  2%. 

Logarithm 
1  erg  or  dyne-centimeter  [dyne-cm]: 

0.001  019  79  gram-centimeter.    Aprx.  V\  ooo 3-008  5096 

0.000  516  328  foot-grain.    Aprx.  3K  -5- 10  000 4-712  9260 

-  0.000  000  1  joule  . 7-000  0000 

-0.000000073761  2  foot-pound.    Aprx.  %•*- 10  000  000..  .  8-8678279 

1  gram-centimeter  [g-cm]: 

=  980.596  6  ergs.    Apr.  1  000 2-991  4904 

-0.000  098  059  66  joule.    Aprx.  Vio  ooo 5-991  4904 

-  0.000  072  330  0  foot-pound.    Aprx.  %  H-  10  000 5-859  3184 

0.000  01  kilogram-meter 5-000  0000 

1  foot-grain  [ft-gr]:  =  1  936.75  ergs.    Aprx.  1900 3-287  0740 

1.975  08  gram-centimeters.  Aprx.  2.  0-295  5836 

-0.000  193  675  joule.    Aprx.  19-4-100  000.  4-287  0740 

-0.000  142  857  ft-lb.    Aprx.  %  -*- 1  000. .  .  .  4-154  9020 
1  joule  [j]  or  volt-coulomb  or  watt-second  [w-sj: 

=       10  000  000.  ergs 7-000  0000 

10  197.9  gram-centimeters.    Aprx.  10  000 4-008  5096 

=         0.737  612  foot-pound.    Aprx.  % 1.867  8279 

-  0.238  882   small  calorie.    Aprx.  24 -h  100 1-378  1834 

=         0.101  979  kilogram-meter.    Aprx.  Vio 1-008  5096 

=   0.001  359  72  metric  hp-second.    Aprx.  "/g  +  l  000 3-133  4483 

^   0.001  341  11   horse-power-second.     Aprx.  %  +  \  000 3-127  4653 

0.001   kilowatt-second.    Aprx.  y\  ooo 3-000  0000 

=  0  000  947  960  thermal  unit.    Aprx.  95  •*•  100  000 3-976  7901 

-0.000526645  pound-Centgr.  heat  unit.    Aprx.  1/19-^  100.   4-7215176 

-0.000  277  778  watt-hour.    Aprx.  «/4i  -4- 1  000 4-443  6975 

-0.000  238  882  large  calorie.    Aprx.  24 -s- 100  000 4-378  1834 

1  foot-pound  [ft-lb]: 

13  557  300.  ergs.    Aprx.  2%  x  1  000  000 7-132  1721 

13  825.5  gram-centimeters.     Aprx.  %  X  10  000..  4-140  6817 

=  1.355  73  joules.    Aprx.Va 0-132  1721 

0.323  859  small  calorie.    Aprx.  *Mo 1-510  3555 

—  0.138  255  kilogram-meter.    Aprx.%o 1-140  6817 

=  0.001  843  40  metric  hp-second.    Aprx.  1Y6  +  1  000. .  .   3-265  6204 

=  0.001  818  18  horse-power-second.    Aprx.  2/n  +  100..   3-259  6373 

=  0.001  355  73  kilowatt-second.     Aprx.%-i-l  000 3-132  1721 

=  0.001  285  17  thermal  unit.    Aprx.  %  +•  1  000 3-1089622 

=         0.000  713  986  Ib-Centgr.  heat  unit.    Aprx.  %  -5- 1  000 .   4-853  6897 
=         0.000  376  591   watt-hour.    Aprx.  %-%- 1000 4-5758696 

—  0.000323859  large  calorie.    Aprx.  ^-^  10  000 2-5103555 

—  0.000000505051   horse-power-hour.  Aprx.  ^^-1  000  000  7-7033348 
=  0.000  000  376  591  kilowatt-hour.    Aprx.  Y%  -f- 1  000  000. .  .   7-575  8696 


ENERGY;  WORK;  HEAT.  75 

1  meter-kilogram:   see  kilogram-meter;  also  under  torque,  below. 

1  kilogram-meter  [kg-m]: 

=            98  059  660.  ergs.    Aprx.  100  000  000 7-991  4904 

=                  100  000.  gram-centimeters ........  5-000  0000 

=              9.805  966  joules.    Aprx.  10 0-991  4904 

=                7.233  00   foot-pounds.    Aprx.  8(Hi 0-859  3184 

=                 2.342  47   small  calories.    Aprx.  % 0-369  6738 

=           0.0133333  metric  hp-second.    Aprx.  %  + 100 2-1249387 

=           0.0131509   horse-power-second.    Aprx.  %  -*•  100 ....  2-1189557 

=       0.009  805  966   kilowatt-second.    Aprx.  Vioo 3-9914904 

=         0.00929567   thermal  unit.    Aprx.  2%  -s- 1  000 3-9682805 

=         0.00516426   Ib-Centgr.  heat  unit.    Aprx.  52 -h  10  000.  3-7130080 

=         0.002  723  88  watt-hour.    Aprx.  %i  -=- 100 3-435  1879 

=         0.002  342  47  large  calorie.    Aprx.  %-t-l  000 3-369  6738 

=  0.000  003  703  70  metric  hp-hour.    Aprx.  ^-r-100  000 6-568  6362 

=  0.00000272388  kilowatt-hour.    Aprx.  3/n-=- 100  000 6-4351879 

1  metric  horse-power-second  [hp-s]: 

=     735.447   joules.    Aprx.^Xl  000 2-866  5517 

=     542.475   foot-pounds.    Aprx.  ^ X  100 2-734  3797 

=     175.685   small  calories.    Aprx.  %  X  100 2-244  7351 

=              75.  kilogram-meters.    Aprx.MXlOO 1-8750613 

=  0.204291    watt-hour.    Aprx.  204 -hi  000 1-3102492 

1  horse-power-second  [hp-s]: 

=     745.650   joules.    Aprx.  %X  1  000 2-872  5348 

550.  foot-pounds  or  11A  X  100 2-740  3627 

=     178.122   small  calories.    Aprx.  %X  100 2-250  7182 

=   76.0404   kilogram  meters.    Aprx.  MX100 1-8810444 

=  0.207  125   watt-hour.    Aprx.  207 -hi  000 1-316  2323 

1  kilowatt-second  [kw-s]: 

1  000.  joules 3-000  0000 

=     737.612   foot-pounds.    Aprx.  MX  1  000 2-867  8279 

=     238.882   small  calories.    Aprx.  240 2-378  1834 

=     101.979   kilogram-meters.    Aprx.  100 2-008  5096 

=  0.277  778   watt-hour.    Aprx.  % 1-443  6975 

1  calorie  (small)  [cal]  or  gram-Centigrade  heat  unit  [g-C]: 

=  0.001  large  calorie  or  kg-Centigr.  heat  unit,  which  see. .  .  .  3-000  0000 

1  thermal  unit  [BTU]  or  pound-Fahrenheit  heat  unitf  [lb-F]: 

1  054.90  joules.    Aprx.  2^ X  100 3-023  2099 

=            778.104  foot-pounds.    Aprx.  %  X  1  000 2-891  0379 

=            251.996  small  calories.    Aprx.  MX1  000 2-4013933 

=            107.577  kilogram-meters.    Aprx.  108 2-031  7195 

=           1.434  36  metric  horse-power-seconds.    Aprx.  ityr....  0-1566582 

=           1.414  74  horse-power-seconds.    Aprx.  10/j- 0-150  6752 


1.054  90  kilowatt-seconds.  Aprx.  add  J^ 
0.555  556  pound-Centgr.  heat  unit.  Aprx. 
0.293  027  watt-hour.  Aprx. 


. 
=     0.251  995  8  large  calorie.    Aprx.  M  ............. 

=  0.000  398  433  metric  hp-hour.    Aprx.  4  -f-  10  000  ---- 

=  0.000  392  982  horse-power-hour.    Aprx.  4^-10  000.  , 
=  0  000  293  027  kilowatt-hour.     Aprx.  8%+  100  000. 


-023  2099 
-744  7275 
-466  9074 
-401  3933 
-600  3557 
-5943727 
-466  9074 


1  pound-Centigrade  heat  unit  [lb-C]: 

=  1  898.81  joules.    Aprx.  1  900 3-278  4824 

=  1  400.59  foot-pounds.    Aprx.  1  400 3-146  3104 

=         453.592  4  small  calories.    Aprx.  %  X  100 2-656  6658 

193.639  kilogram-meters.    Aprx.  194 2-286  9920 

=  2.581  85  metric  horse-power-seconds.  Aprx.  8Vi2-.  -  .  0-411  9307 

=  2.546  52  horse-power-seconds.    Aprx.  *% 0-405  9477 

=  1.898  81  kilowatt-seconds.    Aprx.  i»Ao 0-278  4824 

=  1.800  00  thermal  units.     Aprx.  % 0-255  2725 

=         0.527  448  watt-hour.    Aprx.  10/19  .  / 1-722  1799 

=     0.453  592  4  large  calorie.    Aprx.  10/22 1-656  6658 

=  0.000  717  180  metric  hp-hour.    Aprx.  %  •*- 1  000 4-855  6282 

=  0.000  707  368  horse-power-hour.    Aprx.  5/r  •*•  1  000 4-849  6452 

=  0.000  527  448  kilowatt-hour.    Aprx.  Vio  -s- 100 5-722  1799 

t  Often  called  a  British  Thermal  Unit.     BTU  also  means  kilowatt-hour. 


76  ENERGY;  WORK;   HEAT. 

t  watt-hour  [w-h]: 

3  600.  joules 3-556  3025 

=  2  655.40  foot-pounds.  Aprx.  %  X  1  000 * 3.424  1305 

=  859.975  small  calories.  Aprx.  %  X  1  000 2-934  4859 

=  367.123  kilogram-meters.  Aprx.  ^X  100 2-564  8121 

=  4.89498  metric  horse-power-seconds.  Aprx.  4%o- -  .  0-6897508 

=  4.828  01  hprse-power-seconds.  Aprx.  4^o 0-683  7678 

=  3.6  kilowatt-seconds.  Aprx.  36/io  or  4%i 0-5563025 

=  3.412  66  thermal  units.  Aprx.  8*/10  or  1% 0-533  0926 

=  1.895  92  pound-Centigr.  heat  units.  Aprx.  ™/io 0-277  8201 

=  0.859  975  large  calorie.  Aprx.  % 1-934  4859 

=  0.001  359  72  metric  horse-power-hour.  Aprx.%-M  000..  3.133  4433 

=  0.001  341  11  horse-power-hour.  Aprx.  %  +  l  000 3-127  4652 

0.001  kilowatt-hour 3-000  0000 

1  calorie  (large)  [Cal]  or  kilogram- Centigrade  heat  unit  [kg-CJ: 

=         4  186.17   joules.    Aprx.  4200 3-621  8166 

=         3  087.77  foot-pounds.    Aprx.  3  100 3-489  6446 

=  1  000.  small  calories 3-000  0000 

=       426.9OO   kilogram-meters.    Aprx.  %X  1  000 2-630  3262 

5.692  metric  horse-power  seconds.    Aprx.4% 0-7552649 

=         5.61412  horse-power-seconds.    Aprx.40/4 0-7492819 

=         4.186  17  kilowatt-seconds.    Aprx.  *2Ao 0-621  8166 

=         3.968  32  thermal  units.    Aprx.  4 0-598  6067 

=         2.20462  pound-Cent gr.  heat  units.    Aprx.  22/10 0-3433342 

=         1.162  82  watt-hours.    Aprx.  % -.   0-065  5141 

=  0.001  581  11   metric  horse-power-hour.    Aprx.  %  -5- 1  000.    3-198  9624 

=  0.001  559  48  horse-power-hour.    Aprx.  1^7  -*- 1  000 3-1929794 

=  0.001  162  82  kilowatt-hour.    Aprx.  %-i-l  000 3-065  5141 

1  mile-pound  [ml-lb]: 

5  280.  foot-pounds.    Aprx.  5  300 3-722  6339 

=  0.002  703  66  metric  horse-power-hour.    Aprx.  %-^-l  000  .  3-431  9518 

=  0.002  666  67  horse-power-hour.    Aprx.  %-s-  1  000 3.425  9687 

=  0.001  988  40  kilowatt-hour.    Aprx.  %  ooo 3-298  5035 

1  kilogram -kilometer  [kg-km]: 

=         7  233.00  foot-pounds.    Aprx.  5A  X  10  000 3-859  3184 

1  000.  kilogram-meters 3-000  0000 

=         1.369  89  mile-pounds.    Aprx.  Iy8 0-136  6845 

=  0.003  703  70  metric  horse-power-hour.     Aprx.  %-T-lOO.  .   3-568  6362 

=  0.003  653  03  horse-power-hour.    Aprx.  l%  +  1  000 3-562  6532 

=  0.002  723  88  kilowatt-hour.    Aprx.  ^^l  000 3-435  1879 

1  metric  horse-power-minute  [hp-m]: 

=  44  126.8  joules.    Aprx.  %  X  100  000 4-644  7029 

=  32  548.5  foot-pounds.    Aprx.  1%X  10  000 4-512  5309 

=       4  500.  kilogram-meters.    Aprx.  %  X  1  000 3-653  2125 

=  12.257  5  watt-hours.    Aprx.  «% 1-088  4004 

=  10.541  1   large  calories.    Aprx.  2^ 1-022  8863 

1  horse-power-minute  [hp-m]: 

=  44739.0  joules.    Aprx.  %  X  100  000 4-6506860 

=  33  OOO.  foot-pounds.    Aprx.  MX  100  000 4-518  5139 

=  4  562.42  kilogram-meters.    Aprx.  %  X  1  000 3-659  1956 

=  12.427  5  watt-hours.    Aprx.  «*% L094  3835 

=  10.687  3  large  calories.    Aprx.  1°?^ L028  8694 

1  kilowatt-minute  [kw-m]: 

=    60  000.  joules 4-778  1513 

=  44  256.7  foot-pounds.    Aprx.  %  X  100  000 4-645  9793 

=  6  118.72  kilogram-meters.    Aprx.  6  000 3-786  6609 

=  16.666  7  watt-hours.    Aprx.  K  X  100 1-221  8488 

'  =  14.332  9  large  calories.    Aprx.  1/7  X  100 1-156  3347 


ENERGY;  WORK;  HEAT. 


77 


1  metric  horse-power-hour  [hp-h]: 

=  2  647  610.  joules.    Aprx.   %  X  1  000  000 6-422  8542 

=  1  952  910.  foot-pounds.    Aprx.  %i  X  100  000  000 6-290  6822 

=     270  000.  kilogram-meters.    Aprx.  27  X  10  000 5. 431  3638 

=         3  600.  metric  horse-power-seconds 3-556  3025 

=   2  509.83  thermal  units.    Aprx.  MX  10  000 3-399  6443 

=    1  394.35  pound-Centgr.  heat  units.    Aprx.  1  400 3-144  3718 

=     735.447  watt-hours.   Aprx.  740 2-866  5517 

=     632.467  large  calories.    Aprx.  630 2-801  0376 

=  270.  kilogram-kilometers 2-431  3638 

=•=  60.  metric  horse-power-minutes 1-778  1513 

=  0.986  318  horse-power-hour.    Aprx.  1 1-994  0170 

=  0.735  447  kilowatt-hour.    Aprx.  M  or  2%0 1-866  5517 

1  horse-power-hour  [hp-h]: 

=  2  684  340.  joules.    Aprx.  %  X  1  000  000 6-428  8373 

=  1  980  000.  foot-pounds.    Aprx.  2  000  000 6-296  6652 

=     273  745.  kilogram-meters.    Aprx.  *MX  100  000 5-437  3469 

=         3  600.  horse-power-seconds 3-556  3025 

=     2544.65  thermal  units.    Aprx.  MX  10  000 3-405  6274 

=    1  413.69  pound-Centgr.  heat  units.    Aprx.  1  400 3-150  3549 

«=     745.650  watt-hours.  Aprx.  MX  1  000 2-872  5348 

=     641.240  large  calories.    Aprx.  640 2-807  0207 

=     375.000  mile-pounds  or  s/8  X  1  000 2-574  0313 

60.  horse-power-minutes 1-778  1513 

=    1.013  87  metric  horse-power-hours.    Aprx.  1 0-005  9830 

=  0.745  650  kilowatt-hour.    Aprx.  M 1-872  5348 

1  kilowatt-hour  [kw-h]  [BTUJf: 

=  3  600  000.  joules 6-556  3025 

=  2  655  403.  foot-pounds.    Aprx.  %  X  1  000  000 6-424  1305 

=     367  123.  kilogram-meters.    Aprx.  ^X  100  000 5-564  8121 

=   4  828.01   horse-power-seconds.    Aprx.  4  800 3-683  7678 

=    3  412.66  thermal  units.    Aprx.  3  400 3-533  0926 

=    1  895.92  pound-Centgr.  heat  units.    Aprx.  1  900 3-277  8201 

=         1  000.  watt-hours 3-000  0000 

=     859.975  large  calories.    Aprx.  %  X  1  000 2-934  4859 

=     502.917  mile-pounds.    Aprx.  ^X  1  000 2-701  4966 

=     367.123  kilogram-kilometers.    Aprx.  ^X  100 2-564  8121 

=  60.  kilowatt-minutes 1-778  1513 

=    1.359  72  metric  horse-power-hours.    Aprx.  add  M 0-133  4483 

=    1.341  11   horse-power-hours.    Aprx.  add  l/i 0-127  4652 

Conversion  Tables  for  Energy,   Work,   Heat. 


Foot-lbs.= 

kg-mets 

thermal  u 

Klgr~met'  s 

ft-lbs 

calories 

Thermal  u 

ft-lbs. 

(large) 

Calories  (1) 

kg-mt 

Calories  (s) 

joules 

Joules      — 

calories 

(small) 

1 

0.13826 

7.2330 

0.0012852 

778.10 

0.002  342  5 

426.90 

0.238  88 

4.1862 

2 

0.27G51 

14.466 

0.002  570  3 

1  556.2 

0.004  684  9 

853.80 

0.477  76 

8.3723 

3 

0.41477 

21.690 

0.003  855  5 

2  334.3 

0.007  027  4 

1  280.7 

0.71665 

12.559 

4 

0.553  02 

28.932 

0.0051407 

3  112.4 

0.009  369  9 

1  707.6 

0.955  53 

16.745 

5 

0.691  28 

36.165 

0.006  425  9 

3890.5 

0.011712 

2  134.5 

1.1944 

20.931 

6 

0.829  53 

43.398 

0.0077110 

4  668.6 

0.014055 

2561.4 

1.4333 

25.117 

7 

0.967  79 

50.631 

0.008  996  2 

5446.7 

0.016397 

2988.3 

1.6722 

29.303 

8 

1.1060 

57.864 

0.010281 

6  224.8 

0.018740 

3  415.2 

1.911  1 

33.489 

9 

1.2443 

65.097 

0.011  567 

7  002.9 

0.021  082 

3842.1 

2.1499 

37.676 

10 

1.3826 

72.330 

0.012852 

7781.0 

0.023  425 

4269.0 

2.3888 

41.862 

sat  Britain  this  is  often  called  a  Board  of  Trade  Unit,  or  simply 
id  is  abbreviated  to  BTU;  these  letters  also  stand  for  a 'British 
Jnit,  which  has  an  entirely  different  value. 


t  In  Great  Britain  thi 
a  Unit,  anc1  " 
Thermal  U: 


78  TORQUE. — TRACTIVE     FORCE. 

RELATIONS   BETWEEN   TORQUE   AND   ENERGY. 

Let  foot-pounds,  kilogram -meters,  etc.,  represent  units  of  energy, 
and  let  pound-feet,  meter-kilograms,  etc.,  represent 'units  of  torque; 
a  radian  is  the  angle  whose  arc  is  equal  to  its  radius  (about  57^°);  torque 
acting  through  an  angle  gives  energy;  the  general  relations  between  the 
units  then  are: 

units  of  energy  =  units  of  torque  X  radians ; 

units  of  torque  =  units  of  energy  -f-  radians. 
The  numerical  relations  between  the  units  bearing  similar  names  are: 

1  foot-pound  [ft-lb]  1  pound-foot-radian ; 

1  pound-foot  [lb-ft]  1  foot-pound  per  radian; 

1  foot-pound  =0.159  155  pound-foot-revolution; 

1  pound-foot  =   6.283  19  foot-pounds  per  revolution; 

1  foot-pound  per  revolution  =  0.1 59  155  pound-foot; 

1  pouiid-foot-revolution        =   6.283  19  foot-pounds. 

The  same  relations  are  true  between  kilogram-meters  of  energy  and 
meter-kilograms  of  torque,  or  between  any  other  pairs  of  units  having 
similar  names. 

TRACTION   ENERGY. 

Ton-mile.  A  unit  used  in  traction  calculations  representing  the  energy 
(work,  not  power)  which  is  required  to  draw  one  (short)  ton  of  2  000  Ibs. 
over  a  distance  of  one  mile.  It  has  no  fixed  value,  being  dependent  upon 
the  nature  of  the  track,  the  grade,  and  the  speed.  It  is  never  used  simi- 
larly to  the  term  "foot-pound"  as  representing  one  ton  raised  one  mile 
vertically.  See  also  note  on  "pound  per  ton"  under  tractive  effort, 
below. 

Ton-kilometer.  A  unit  similar  to  "ton-mile,"  but  meaning  one  metric 
ton  drawn  over  a  distance  of  one  kilometer. 

Ton  per  mile.  A  term  popularly  (though  not  correctly)  used  for  "ton- 
mile"  (see  above).  The  term  "per"  is  here  used  incorrectly,  as  in  all 
other  cases  it  means  that  the  first  quantity  is  divided  by  the  second,  and 
not  multiplied,  as  in  this  case. 

Car-mile.  A  term  used  in  traction  calculations  representing  the  energy 
required  to  draw  a  car  one  mile.  It  is  analogous  to  ton-mile  (see  above), 
but  is  even  less  fixed  in  value,  as  it  also  involves  the  weight  of  the  car. 

The  only  fixed  relations  which  these  units  have  are  the  following: 

Logarithm 

1  ton-kilometer  [t -km ]==  0.684  943  ton-mile.    Aprx.  %3 1-835  6545 

1  ton-mile  [tn-ml]  =1.459  98  ton-kilometer.    Aprx.  18/0..  .   0-164  3455 

(Aprx.  means  within  2#.) 

TRACTIVE    FORCE;     TRACTIVE     EFFORT;     TRAC- 
TION   RESISTANCE;    TRACTION    COEFFICIENT. 

(Force  -f-  weight.) 

The  following  units,  although  really  of  the  nature  of  forces,  have  been 
placed  here  in  order  to  accompany  traction  energy. 

Pound  per  ton.  A  unit  used  in  traction  calculations,  representing  the 
force  in  pounds  required  to  move  one  (short)  ton  of  2  000  pounds  hori- 
zontally against  the  friction  of  the  rails,  wheels,  roads,  etc.  It  has  no  fixed 
value,  as  it  varies  with  this  friction.  It  is  really  a  mere  coefficient,  rela- 
tion, or  mere  number,  and  has  no  dimensional  formula.  See  also  note  on 
"ton-mile"  under  Energy  units. 

Kilogram  per  ton.  A  unit  similar  to  "pound  per  ton,"  but  meaning 
the  force  in  kilograms  required  to  draw  one  metric  ton. 

The  only  fixed  relations  which  these  units  have  are  the  following: 

Logarithm 

1  pound  per  (short)  ton  [lb/tn]:    =  0.500  000  kg  per  (met.)  tn.   1.698  9700 
1  kilogram  per  (met.)  ton  [kg/t]  =  2.000  000  Ib  per  (short)  ton.  0-301  0300 


POWER.  79 

POWER;  RATE  of  ENERGY;  RATE  of  DOING 
WORK  j  MOMENTUM.  (Energy  -f-  time  j  mass  X 
velocity.) 

Units  of  power  are  for  measuring  the  rate  of  doing  work,  and  should 
therefore  be  clearly  distinguished  from  the  units  of  work  or  heat,  which 
are  energy  and  not  power.  Much  confusion  is  often  caused  by  confound- 
ing these  two  terms  with  each  other.  Power  bears  the  same  relation  to 
energy  (work,  heat,  etc.)  as  a  velocity  does  to  length;  power  is  energy 
divided  by  time,  just  as  velocity  is  length  divided  by  time.  A  reduction 
from  power  to  energy  or  the  reverse  therefore  always  involves  the  factor 
of  time. 

There  are  really  only  two  true  power  units  in  common  use — the  horse- 
power and  the  watt  (or  kilowatt) — but  powers  are  also  often  expressed  in 
energy  per  unit  of  time,  as  in  foot-pounds  per  minute.  This  table  of  re- 
duction factors  is  confined  in  general  to  the  true  power  units.  A  large 
number  of  reduction  factors  in  terms  of  units  of  energy  have,  however, 
also  been  included,  but  these  are  in  general  given  here  only  per  minute  and 
not  also  per  second  and  per  hour,  as  this  would  have  made  the  table  many 
times  as  long  and  very  cumbersome  to  use ;  a  mere  multiplication  or  divi- 
sion by  60  will  then  reduce  the  energy  per  minute  to  its  equivalent  in  energy 
per'hour  or  per  second,  respectively. 


avoid  all  such  confusion: 

If  W  is  any  unit  of  energy  (such  as  work  or  heat)  like  a  foot-pound,  heat 
unit,  etc.,  then 

1  W  per  hour      =^o  W  per  minute; 

=  \i  coo  W  per  second ; 
1  W  per  minute  =  60  W  per  hour; 

~y&o  W  per  second; 

1  W  per  second  =  3  600  W  per  hour; 

=  60  W  per  minute. 

Thus  120.  ft-lbs  per  min  =  (120X60)  =  7  200.  ft-lbs  per  hour  or  (120^-60) 
=  2.  ft-lbs  per  sec. 

For  reducing  powers  which  are  expressed  in  energy  units  per  minute  (like 
ft-lbs  per  min)  to  powers  expressed  in  other  energy  units,  but  also  per  min- 
ute (like  heat  units  per  min)  use  the  table  for  the  energy  units  (ft-lbs  into 
heat  units  in  this  case);  the  element  of  time  then  does  not  enter,  as  it  is  the 
same  in  both.  If  one  is  per  minute  and  the  other  per  second,  they  must, 
of  course,  both  be  first  reduced  to  either  minutes  or  seconds. 

Some  of  these  same  units  also  measure  momentum,  only  that  they 
then  mean  masses  multiplied  by  velocities.  In  the  units  of  power  the 
pounds,  kilograms,  etc.,  represent  forces,  while  in  the  units  of  momentum 
they  represent  masses. 

Power  factor  is  a  term  used  to  show  the  amount  of  true  power  con- 
tained in  a  given  amount  of  apparent  power.  It  is  the  ratio  of  the  true 
power  to  the  apparent  power.  Its  use  is  limited  chiefly  to  electric  power 
generated  by  alternating  currents.  With  direct  electric  currents  the  power 
is  equal  to  the  product  of  the  volts  and  the  amperes,  and  is  called  watts; 
with  alternating  currents,  however,  this  is  true  only  when  the  volts  and 
amperes  are  exactly  in  phase  with  each  other,  which  often  is  not  the  case. 
When  there  is  such  a  difference  in  phase,  that  is,  when  the  current  lags 
behind  or  precedes  the  voltage,  their  product  is  only  apparent  power  and 
is  usually  measured  in  volt-amperes.  If  the  true  power  in  such  a  case  is 
measured  in  watts,  then  the  power  factor  will  be  the  number  of  watts 
divided  by  the  number  of  volt-amperes,  and  it  will  always  be  less  than 
unity,  in  practice  usually  between  about  0.7  and  0.95.  For  true  sine  waves 
the  real  power  in  watts  is  equal  to  the  voltage X current  X cos  A,  in  which 
0  is  the  angular  phase  difference;  hence  it  follows  that  in  such  cases  the 
power  factor  is  numerically  equal  to  cos  <J>,  which  is  found  directly  from  a 
table  of  cosines.  Sometimes  the  power  factor  is  stated  in  percent, in  which 
case  it  is  equal  to  the  above  figure  multiplied  by  100. 


80  POWEK. 


Load  factor  is  a  term  commonly  applied  to  electric,  steam,  or  hydraulic 
ower  stations  to  show  how  much  of  the  total  possible  amount  of  power 
as  actually  been  generated  or  used  during  a  limited  time,     It  is  the  ratio 
of  the  mean  power  used  during  a  limited  time  (generally  1  day)  divided  by 


p 
h 


the  total  power  that  the  station  could  have  generated  during  that  time; 
as  it  is  usually  stated  in  percent,  this  ratio  must  be  multiplied  by  100.  If 
the  average  power  generated  during  a  day  is  %  of  that  which  the  station 
is  capable  of  generating,  the  load  factor  is  25%.  A  100%  load  factor 
means  that  the  station  is  running  at  its  full  output  all  the  time.  In  water- 
power  installations  or  in  stations  having  storage  batteries,  this  quantity 
is  of  use  in  determining  the  amount  of  storage  capacity  desired. 

POWER;     RATE      of     ENERGY;      RATE     of     DOING 
WORK  ;    MOMENTUM. 

Aprx.  means  within  2%. 

Logarithm 
1  erg  per  second  or  1  dyne-centimeter  per  second; 

=  0.000  000  1  watt  ....................................   7-000  0000 

1  grain-centimeter  per  second  [g-cm/s]: 

=  0.000  098  059  7  watt.    Aprx.  ^o  ooo  ....................   5-991  4904 

1  foot-grain  per  second  [ft-gr/s]: 

=  0.000  193  675  watt.    Aprx.  %i  -4-  100  ...................  4-287  0740 

1  foot-pound  per  minute  [ft-lb/min]: 

=         0.022  595  4  watt.    Aprx.  9/i  -4-  100  ..................   2-3540208 

=         0.011  363  6  mile-pound  per  hour.    Aprx.  8A  -4-  100  .....   5-055  5174 

=  0.000  030  723  4  metric  horse-power.    Aprx.  Vis  •*•  10  000.  .    5-487  4691 
=  0.000  030  303  0  horse-power.    Aprx.  3  -4-  100  000  .........    5-481  4860 

=  0.000  022  595  4  kilowatt.    Aprx.  %  -s-  100  000  ...........   5-354  0208 

1  calorie  (small)  per  minute  [cal/min]: 

=  0.069'769  5  watt.    Aprx.  Vioo  .............  '.  ...........   2-843  6653 

1  kilogram-meter  per  minute  [kg-m/min]: 

=         0.163  433  watt.    Aprx.  Y&  .........................   1.213  3391 

=      0.082  193  2  mile-pound  per  hour.    Aprx.  %  -4-  10  .......   2-914  8357 

=  0.000  222  222  metric  horse-power.    Aprx.  %  -4-  1  000  ......   4-346  7875 

=  0.000  219  182  horse-power.    Aprx.  %  -4-  1  000  ............   4-340  8044 

-0.000  163  433  kilowatt.    Aprx.K-J-1  000  ...............   4-213  3391 

1  watt  [w]  or  1  joule  per  second: 

=  10  000  000-  ergs  per  second  .........................   7-000  0000 

=         10197.9  gram-centimeters  per  second.  Aprx.  10000.   4-0085096 
=         5  163.28  foot-grains  per  second.    Aprx.  5  200  .......   3-7129260 

=         44.256  7  foot-pounds  per  minute.    Aprx.  %  X  100  ____    1.645  9793 

=         14.332  9  small  calories  per  minute.    Aprx.  Vr  X  100.  .  .   1-156  3347 
=         6.118  72  kilogram-meters  per  minute.    Aprx.  6  ......   0-786  6609 

=       0.737  612  foot-pound  per  second.    Aprx.  %  ..........   1-887  8279 

=       0.502  917  mile-pound  per  hour.    Aprx.  H  ............   1-7014965 

=       0.238  882  small  calorie  per  second.    Aprx.  24-4-100.  .  .    1-378  1834 
=       0.101979  kilogram-meter  per  second.    Aprx.^o  ......    1-0085096 

=   0.056  877  6  thermal  unit  per  minute.    Aprx.  #-4-10  ____    2-754  9414 

=   0.0315987  Ib-Centgr.  heat  unit  per  min.  Aprx.  2%^-  100  2-4996689 
=   0.014  332  9  large  calorie  per  minute.    Aprx.  ^70  ........    2-156  3347 

=  0.001  359  72  metric  horse-power.    Aprx.  %  •*•  1  000  ......   3-133  4483 

=  0.001  341  11   horse-power.    Aprx.  %  -4-1  000  ............   3-127  4653 

0.001   kilowatt  ...............................   3-000  0000 

•watts  =  volt-amperes  X  cos  angle  of  lag. 

volt-amperes  =  volts  X  amperes. 

=  watts  -4-  cos  angle  of  lag. 

1  foot-pound  per  second  [ft-lb/s]: 

60.  foot-pounds  per  minute  ..................   1-778  1513 

=         8.29532   kilogram-meters  per  minute.    Aprx.  %X  10.  0-9188330 
=         1.355  73  watts.    Aprx.  %  ........................   0-132  1721 

=  0.001  843  40  metric  horse-power.    Aprx.  *%  -4-  1  000  .....   3-265  6204 

0.001  818  18  horse-power.    Aprx.  34i-^lOO  .............   3-259  6373 

355  73  kilowatt.    Aprx.  %-s-  1  000  ...............  3-182  1721 


,  POWER.  81 

1  mile-pound  per  hour 

88.  foot-pounds  per  minute.    Aprx.  %  X  100 1 .944  4827 

=         12.1665  kilogram-meters  per  minute.    Aprx.  12 1-0851644 

=         1.988  40   watts.    Aprx.  2 0-298  5035 

=  0.002  666  67   horse-power.    Aprx.  8/3^- 1  000 3-425  9687 

1  calorie  (small)  per  second  [eal/s]  =  4.186  17  watts.    Ap.  e%2.  0-621  8166 
1  kilogram-meter  per  second  [kg-m/s]: 

—          433.980   foot-pounds  per  minute.    Aprx.  %  X  1  000 .    2-6374696 

=  60.  kilogram-meters  per  minute 1-778  1513 

=         9.805  97   watts.    Aprx.  10 0-991  4904 

=   0.0133333  metric  horse-power.    Aprx.  %-*•  100 2-1249387 

=   0.013  150  9  horse-power.    Aprx.  4/3  -*- 100 2-118  9557 

=  0.009  805  97   kilowatt.    Aprx.  1-4-100 3-991  4904 

1  thermal  unit  per  minute  [lb-F/minl: 

=       17.581  6  watts.    Aprx.  %  X  10.  /. 1-245  0586 

=  0.023  906  0  metric  horse-power.  Aprx.  1%-f- 100 1-378  5069 

=  0.023  578  9  horse-power.    Aprx.  %-^-100 2-372  5239 

=  0.017  581  6  kilowatt.    Aprx.  %  +  WQ 2-245  0586 

1  pound-Centigrade  heat  unit  per  minute  [lb-C/min]: 

=       31.646  9  watts.    Aprx.  32 1.500  3311 

=  0.043  030  8  metric  horse-power.    Aprx.  %  -*- 10 2-633  7794 

=  0.042  442  1  horse-power.    Aprx.  %  -*•  10    2-627  7964 

'    =0.031  646  9  kilowatt.    Aprx.  32-r-l  000 2-500  3311 

1  watt-hour  per  minute  =60.  watts 1-778  1513 

1  calorie  (large)  per  minute  [Cal/min]: 

=       69.769  5  watts.    Aprx.  70 1-843  6653 

=  0.094  866  7  metric  horse-power.    Aprx.  %i 2-977  1136 

=  0.093  568  7  horse-power.    Aprx.  %2 2-971  1306 

=  0.069  769  5  kilowatt.    Aprx.  Vioo 2-843  6653 

1  mile-pound  per  minute  [ml-lb/min]: 

=     119.304  watts.    Aprx.  120 2-0766548 

=  0.162  220  metric  horse-power.    Aprx.  %n-10 1.210  1031 

=  0.160  000  horse-power.    Aprx.  %-f- 10 1-204  1200 

-0.119  304  kilowatt.    Aprx.  %-r-lO.  .  , 1-076  6541 

1  kilogram -kilometer  per  minute  [kg-km/min]: 

=     163.433  watts.    Aprx.  Y&  X  1  €00 2-213  339] 

=  0.222  222  metric  horse-power.    Aprx.  % 1-346  7875 

=  0.219  182  horse-power.    Aprx.  % 1-340  8044 

=  0.163  433  kilowatt.    Aprx.  % 1-213  3391 

1  metric    horse-power    [hp]    or    French    horse-power  or 
chevalvapeur  or  force  de  cheval  or  Pferde-kraft : 

=  7.354  48 X109  ergs  per  second.    Aprx.  2%X109 9-8665517 

=          32  548.5   foot-pounds  per  minute.    Aprx.  33  000 4-512  5309 

=  4  500.  kilogram-meters  per  minute.  Aprx.  »/2X  1000  3-653  2125 

735.448   watts.    Aprx.  2^X  100 2-866  5517 

=  542.475   foot-pounds  per  second.    Aprx.  °/ii  X  1  000  ..   2-7343797 

=  75.  kilogram-meters  per  second  or  9-4  X  1 00  „ 1-875  0613 

=          41.830  5   thermal  units  per  minute.    Aprx.  42 1-621  4931 

=          23.239  2   Ib-Ctg.  heat  units  per  minute.  Aprx.  %  X  10.  1-366  2206 

=          10.541  1    large  calories  per  minute.    Aprx.  2K 1-022  8864 

=       0.986  318   horse-power.    Aprx.  1 1-994  0170 

=        0-750  000   poncelet 1-875  0613 

=       0.735  448   kilowatt.    Aprx.  2%-^-10.. 1-866  5517 

1  horse-power  [hp]: 

=  7.456  SOX  109  ergs  per  second.    Aprx.  MX  10*° 9-872  5348 

=  33  OOO.  foot-pounds  per  min.  Aprx.  MX  100  000.  .  .   4-518  5139 

=          4562.42   kg-meters  per  minute.    Aprx.  %  XJ.  000 3-6591956 

=  745.650   watts.    Aprx.  MX  1  000 2-872  5348 

«=  550.  foot-pounds  per  second.    Aprx.  1^X100.  .  .   2-740  3627 

=  375.000   mile-pounds  per  hour.    Aprx.  %Xl  000...  .   2-574  0313 

=          76.040  4   kg-meters  per  second.    Aprx.  %X  100 1-881  0444 

=          42.4108   thermal  units  per  min.  Aprx.  3A  X  100 1-6274762 

=          23.5615   Ib-Ctg.  heat  units  per  min.     Aprx.  %  X  10.  .   1-3722037 

=          10.687  3   large  calories  per  min.    Aprx.  3%j 1-028  8695 

=          1.013  87   metric  horse-powers.    Aprx.  1 0-005  9830 

=        0.760  404   poncelet.  Aprx.  % 1-881  0444 

=       0.745  650   kilowatt.  Aprx.  M 1-872  5348 


82 


POWER. 


I  poncelet  =  100.  kilogram -meters  per  second 2-000  0000 

41         =    1.33333   metric  horse-powers.    Aprx.  % 0-1249387 

=    1.315  09    horse-powers.    Aprx.  % : 0-118  9557 

=  0.980  597    kilowatt.    Aprx.  1 L991  4904 

1  kilowatt  [kw]  =    IX  1010  ergs  per  second 10-000  0000 

=  44256.7  ft-lbs  per  min.  Aprx.  %X  100  000..  4-6459793 
=  6118.72  kg-met.  per  min.  Aprx.  6  X  1  000 .  .  3-7866609 

=       1  000.  watts 3-000  0000 

=  737.612  ft-lbs.  per  sec.  Aprx.  MXl  000...  .  2-867  8279 
=  101.979  kilogram-metr.  per  sec.  Aprx.  100.  2-0085096 
=  56.8776  thermal  u.  per  min.  Aprx.  #  X  100.  1-7549414 
=  31.598  7  Ib-Ctg.  heat  u.  per  min.  Aprx.  <*%..  1.499  6689 
=  14.3329  large  cal.  per  min.  Aprx.  14  X  100..  1.1563347 
=  1.35972  metric  horse-powers.  Aprx.  add}^.  0-1334483 

=  1.341  11   horse-powers.    Aprx.  add  1A 0-127  4652 

=  1.019  79  poncelets.    Aprx.  1 0-008  5096 

1  watt-lrour  per  second  =  3  600.  watts.    Aprx.  ^Xl  000.  .  .   3-556  3025 

=  3.600  kilowatts.    Aprx.  1H 0-556  3025 

1  metric  horse-power-hour  per  minute: 

=  60.  metric  horse-powers 1-778  1513 

1  hors«-power-hour  per  minute  =60.  horse-powers 1-778  1513 

1  kilowatt-hour  per  minute  =  60.  kilowatts 1-778  1513 

1  metric  horse-power-hour  per  second: 

=  3  600.  metric  horse-powers.    Aprx.  ^X  1  000 3-556  3025 

1  horse-power-hour  per  second: 

=  3  600.  horse-powers.    Aprx.  ^ XI  000 3-556  3025 

1  kilowatt  hour  per  second: 

=  3600.  kilowatts.    Aprx.  ^X  1000 3-5563025 


Conversion  Tables  for  Power. 


Horse-powers  = 

kilowatts  . 

metrhp';-* 

Meti'ic  hp's      — 

kilowatts 

horse- 

Kilowatts       = 

horse- 
powers 

metrhp's 

powers 

1 
2 
3      . 
4 
5 
6 
7 
8 
9 
10 

0.74565 
1.4913 
2.2370 
2.9826 
3.7283 

4.4739 
5.2196 
5.965  2 
6.7109 
7.456  5 

1.341  1 
2.682  2 
4.023  3 
5.3644 
6.705  6 

8.046  7 
9.387  8 
10.729 
12.070 
13.411 

0.735  45 
1.4709 
2.206  3 
2.941  8 
3.6772 
4.4127 
5.1481 
5.8836 
6.6190 
7.3545 

1.3597 
2.7194 
4.079  2 
5.438  9 
6.7986 
8.1583 
9.5180 
10.878 
12.237 
13.597 

1.0139 
2.027  7 
3.041  6 
4.055  5 
5.069  4 
6.0832 
7.097  1 
8.1110 
9.1248 
10.139 

0.986  32 
1.9726 
2.9590 
3.945  3 
4.9316 
5.9179 
6.9042 
7.890  5 
8.876  9 
9.8632 

FORCES.  83 

FORCES  5    WEIGHTS    Considered    as    Forces.      (See    also 
Weights.)" 

Only  true  units  of  force  (dynes  and  poundals)  are  given  here,  together 
with  their  values  in  terms  of  weights,  and  the  reciprocals  of  these  values. 
The  relations  between  two  weights  when  considered  as  forces  are,  of  course, 
the  same  as  when  they  are  considered  as  masses;  these  have  been  given 
under  Weights  and  are  therefore  not  repeated  here. 

A  true  unit  of  force  is  independent  of  the  value  of  gravity  and  is  the 
same  throughout  the  universe.  But  a  weight  considered  as  a  force  in- 
cludes the  value  of  gravity  and  is  therefore  different  for  different  values 
of  gravity.  The  value  of  gravity  used  in  these  tables  is  980.596  6  (see 
note  on  this  value  under  the  units  of  Acceleration). 

A  dyne  is  that  force  which,  acting  on  a  mass  of  one  gram  for  one  second, 
produces  a  velocity  of  one  centimeter  per  second;  this  definition  refers 
to  a  space  which  is  free  from  the  attraction  of  other  bodies.  A  poundal 
is  similarly  that  force  w^iich,  acting  on  a  mass  of  one  pound  for  one  second, 
produces  a  velocity  of  one  foot  per  second. 

The  attraction  of  gravity  of  the  earth  is  really  a  force,  but  it  cannot  be 
used  as  a  unit  of  force,  as  the  amount  of  this  force  which  acts  on  anybody 
depends  on  the  mass  on  which  it  acts.  When  reduced  to  the  force  per 
gram  mass,  it  becomes  the  same  thing  as  the  force  represented  by  one  gram 
considered  as  a  weight,  and  this  together  with  all  similar  values  is  given 
in  the  table.  The  attraction  of  gravitation  becomes  a  constant  quantity 
when  it  is  stated  as  an  acceleration,  as  in  this  form  it  is  independent  of 
the  mass  on  which  it  acts;  it  can  then  be  used  as  a  unit  and  is  included 
as  such  in  the  table  of  accelerations,  which  see  for  its  reduction  factors. 

For  tractive  forces  or  tractive  efforts,  see  under  this  title  at  the  end  of 
units  of  Energy. 

Aprx.  means  within  2%. 

Logarithm 

1  microdyne  =  0.000  001  dyne 6-000  0000 

1  milligram  =         0.980  596  6  dyne.    Aprx.  98-=-  100 1.991  4904 

=  0.000  070926  5  poundals.    Aprx.  7 -=-100  000.  ..   5-850  8088 

1  dyne  =  1 .019  79  milligrams.    Aprx.  1 0-008  5096 

=  0.015  737  7  grain.    Aprx.  *#  -f-  100 2-196  9418 

=         0.001  019  79  gram.    Aprx.  1  -=- 1  000 3.008  5096 

=   0.000  035  971  9  ounce  (av.).    Aprx.  4 -=-110  000 5-5559637 

=   0.000  072  330  0  poundal.    Aprx.  ^-=-10  000 5.859  3184 

-0  000  002  248  25  pound  (av.).    Aprx.  %  -=- 1  000  000. .  .  .   6-351  8437 

1  grain  =         63.541  6  dynes.    Aprx.  Vii  X  100 1-8030582 

=  0.001  595  96  poundal.    Aprx.  46^-10  000 3-662  3766 

1  gram  =  980.596  6  dynes.    Aprx.  1  000 2-991  4904 

=  0.070926  5  poundal.    Aprx.7-^100 2-8508088 

1  kilodyne  =  1  000.  dynes 3-000  0000 

1  myriadyne  =  10  000.  dynes 4-000  0000 

lpoundal=       14  099.1  milligrams.    Aprx.  ^r  X  100  000 4-1491912 

-       13  825.5  dynes.    Aprx.  *HX  10  000 ...4-1406816 

=        217.582  grains.    Aprx.  220 2-337  6234 

=       14.099  1  grams.    Aprx.  Vi  X  100 1-1491912 

=     0.497  331  ounce  (av.).    Aprx.  ^ 1-696  6453 

=  0.031  083  2  pound.    Aprx.  31-4-1  000 2-492  5253 

=  0.014  099  1  kilogram.    Aprx.  ^o 2-1491912 

1  ounce  (av.)  =  27  799.5  dynes.    Aprx.  ^  X  10  000 4.444  0363 

=  2.010  73  poundals.    Aprx.  2 '.   0-303  3547 

1  pound  (av.)=  444  791.  dynes.    Aprx.%Xl  000000 5-648  1563 

=  32.171  7   poundals.    Aprx.  32 1-507  4746 

I  kilogram  =980  596.6  dynes.    Aprx.  1  000  000 5.991  4904 

==   70.926  5  poundals.    Aprx.  70 1-850  8088 

=  0.980  597  megadyne.    Aprx.  1 1.991  4904 

1  megadyne  =  1  000  000.  dynes 6-000  0000 

=   72.330  0  poundals.    Aprx.  5/<r  X  100 1.859  3184 

=   2.248  25  pounds.    Aprx.  % 0-351  8437 

=    1.019  79  kilograms.    Aprx.  1 0-008  5096 

-d-TractiveForces,  p.  78. 


84  MOMENTS    OF    INERTIA. 


MOMENTS  of  INERTIA.     (Mass  X  square  of  length.) 

The  moment  of  inertia  of  a  body  is  its  mass  multiplied  by  the  square  of 
the  radius  of  gyration;  it  must  therefore  always  refer  to  some  axis  of  rota- 
tion. It  represents  that  weight  which  when  concentrated  at  a  unit  distance 
from  the  axis  of  rotation  would  require  the  same  energy  to  cause  a  given 
increase  in  its  angular  velocity,  that  the  body  itself  requires.  It  bears 
the  same  relation  to  angular  acceleration  as  weight  bears  to  linear  acceler- 
ation. 

Moments  of  inertia  (so-called)  are  frequently  used  in  calculating  the 
strength  of  beams  of  various  cross-sections.  In  such  cases  the  formulas 
usually  represent  the  moments  of  inertia  of  each  of  differently  shaped  cross- 
sections,  and  the  mass  is  then  usually  represented  by  its  cross-sectional 
area,  as  the  formulas  are  then  the  same  for  all  materials  provided  only  that 
the  shape  of  the  cross-section  is  the  same;  the  numbers  thus  obtained  are 
often  (though  riot  correctly)  called  the  moments  of  inertia;  they  might 
better  be  called  the  specific  moment  of  inertia  of  the  respective  cross-section. 
The  figure  obtained  from  such  a  formula  must  then  be  multiplied  by  the 
mass  (weight)  of  a  cube  (of  unit  side)  of  the  material  to  give  the  true  mo- 
ment of  inertia,  if  that  is  required,  of  a  slab  or  section  having  a  thickness 
of  one  unit  of  length.  In  calculations  involving  the  ratio  of  two  differ- 
ent moments  of  inertia,  the  weight  or  mass  need  not  be  introduced  pro- 
vided the  material  is  the  same  in  both;  in  all  such  cases  the  figures  obtained 
from  the  formulas  just  mentioned  can  be  used  directly.  In  such  cases  the 
relation  between  the  units  in  which  the  moments  of  inertia  are  expressed 
is  as  the  fourth  power  of  the  respective  linear  units. 


MOMENTS  of  INERTIA  in  terms  of  the  mass. 

Logarithm 
1  unit  in  pounds  and  indies: 

=  2.926  41  units  in  kilograms  and  centimeters 0-466  3350 

1  unit  in  kilograms  and  centimeters: 

=  0.341  716  unit  in  pounds  and  inches , 1-533  6659 


MOMENTS  of  INERTIA  in  terms  of  the  surface. 

1  unit  in  inches  =  41. 623  5  units  in  centimeters 1-619  o'384 

1  unit  in  centimeters  =  0.024  024  9  unit  in  inches 2-380  0616 

Thus  if  the  moment  of  inertia  (so-called)  of  the  cross-section  of  a  beam, 
for  instance,  has  been  calculated  in  inches  and  square  inches  and  is  then 
multiplied  by  41.62,  the  result  would  be  the  same  as  if  the  moment  of 
inertia  of  the  same  cross-section  had  been  calculated  in  centimeters  and 
square  centimeters. 


MOMENTS  of  MOMENTUM ;  ANGULAR  MOMENTUM, 

(Momentum  X  length.) 

These  units  are  simply  those  of  momentum  multiplied  by  a  length,  and 
as  they  are  seldom  used  it  is  not  necessary  to  give  them  in  a  separate  table. 
The  units  of  momentum  are  given  above  (see  under  Power);  the  length  by 
which  they  are  to  be  multiplied  must  of  course  be  in  terms  of  the  same  unit 
as  the  one  already  included  in  the  respective  unit  of  momentum.^  The 
moment  of  momentum  is  also  equal  to  the  moment  of  inertia  divided  by 
time. 


LINEAR  VELOCITIES;   SPEEDS.  85 

LINEAR  VELOCITIES;  SPEEDS.     (Length H- time.) 

For  simple  reductions  or  relations  between  lengths,  see  units  of  Length. 
Aprx.  means  within  2%. 

Logarithm 

1  foot  per  minute  [ft/min]: 

=     0.304  801  meter  per  minute.    Aprx.  sAo 1-484  0158 

=  0.018288  0  kilometer  per  hour.    Aprx.  2Ai-^-10 2-2621671 

=  0.016  666  7  foot  per  second.    Aprx.  K^-10. 2-221  8487 

=  0.011  363  6  mile  per  hour.    Aprx.  8/<r-=-100 2-055  5174 

1  kine  =  1  centimeter  per  second 0-000  0000 

1  centimeter  per  second  [cm/s]: 

=       1  kine 0-000  0000 

=  0.01  meter  per  second,  which  see  for  other  values 2-000  0000 

1  meter  per  minute  [m/min]: 

=       3.280  83  feet  per  minute.    Aprx.  *% 0-515  9842 

=  0.06  kilometer  per  hour 2-778  1513 

=  0.054  680  6  foot  per  second.    Aprx.  ^-fr-lOO 2-737  8329 

=  0.037  282  2  mile  per  hour.    Aprx.  ^-*-10 2-5715016 

1  kilometer  per  hour  [km/hr]: 

=   54.680  6  feet  per  minute.    Aprx.  *HX10 1-737  8329 

=    16.666  7  meters  per  minute.    Aprx.  KX100 1-2218487 

=  0.911  343  foot  per  second.    Aprx.  i%i 1-959  6816 

=  0.621  370  mile  per  hour.    Aprx.  Y% ...M 1-7933503 

=  0.539611  knot  (Brit.)  per  hour.    Aprx.  <Hi 1-732  0806 

=  0.539  593  knot  (U.  S.)  per  hour.    Aprx.  «/n 1.732  0660 

1  foot  per  second  [ft/s]: 

60.  feet  per  minute 1-778  1513 

=       18.288  0  meters  per  minute.    Aprx.  ^X  10 1-262  1671 

=       1.097  28  kilometers  per  hour.    Aprx.  add  Ho 0-040  3184 

=     0.681  818  mile  per  hour.    Aprx.  68-^-100 1.833  6687 

=     0.592  105  knot  (Brit.)  per  hour.    Aprx.  6/io 1-772  3990 

=     0.592  085   knot  (U.  S.)  per  hour.    Aprx.  6/40 1-772  3844 

=     0.304  801   meter  per  second.    Aprx.  %o 1-484  0158 

=  0.018  288  0  kilometer  per  minute.    Aprx.  ^-s-lOO 2-262  1671 

=  0.011  363  6  mile  per  minute.    Aprx.  8/7-^100 2-055  5174 

1  mile  per  hour  [ml/hr]: 

88.  feet  per  minute 1-944  4827 

=       26.822  4  meters  per  minute.    Aprx.  %  X  10 1.428  4984 

=       1.609  35  kilometers  per  hour.    Aprx.  % 0-206  6497 

=       1.466  67  feet  per  second.    Aprx.  *% 0-166  3313 

=     0.868421   knot  (Brit.)  per  hour.    Aprx.  % 1-9387303 

=     0.868  392   knot  (U.  S.)  per  hour.    Aprx.  % 1.938  7157 

=     0.447041   meter  per  second.    Aprx.  %-^  10 1-6503471 

=  0.026  822  4  kilometer  per  minute.    Aprx.  %  -*- 100 2-428  4984 

=  0.016  666  7  mile  per  minute.    Aprx.  K^-10 2-221  8487 

1  knot  (Brit.)  per  hour  : 

=    1.853  19  kilometers  per  hour.    Aprx.  ^ 0-267  9194 

=    1.688  89  feet  per  second.    Aprx.  1% 0-227  6011 

=    1.151  52  miles  per  hour.    Aprx.  add  Vi 0-061  2697 

=  0.999  966  knot  (U.  S.)  per  hour.    Aprx.  1 1. 999  9354 

1  knot  (U.  S.)  per  hour  : 

=    1.853  25  kilometers  per  hour.    Aprx.  ^ 0-267  9340 

=    1.688  94  feet  per  second.    Aprx.  *% 0-227  6157 

=    1.151  55  miles  per  hour.    Aprx.  add  VT 0-061  2843 

=  1.000  034  knots  (Brit.)  per  hour.    Aprx.  1 0-000  0146 

1  meter  per  second  [m/s]: 

=        196.850  feet  per  minute.    Aprx.  200 2-294  1355 

100.  centimeters  per  second 2-000  0000 

60.  meters  per  minute 1-778  1513 

3.6   kilometers  per  hour 0-556  3025 

=       3.280  83  feet  per  second.    Aprx.  *% 0-515  9842 

2.236  93  miles  per  hour.    Aprx.  % 0-349  6529 

006   kilometer  per  minute 2-7781513 

=  0.037  282  2  mile  per  minute.    Aprx.  >£7 2-571  5016 


86      ANGULAR  VELOCITIES;    ROTARY  SPEEDS. 

1  kilometer  per  minute  [km/min]: 

=   54.680  6  feet  per  second.    Aprx.  55 1.737  8329 

=   37.282  2  miles  per  hour.    Aprx.  37 , 1-571  5016 

=    16.666  7  meters  per  second.    Aprx.  HX  100 1-221  8487 

=  0.621  370  mile  per  minute.    Aprx.  y% 1.793  3503 

1  mile  per  minute  [ml/min]: 

88.  feet  per  second , 1-944  4827 

60.  miles  per  hour 1-778  1513 

=  26.822  4  meters  per  second.    Aprx.  27 1-428  4984 

=  1.609  35  kilometers  per  minute.    Aprx.  % 0-206  6497 

Miscellaneous  concrete  units:  Average  velocity  of  molecules 
about  500.  meters  per  second  (Woodward).  Velocity  of 
light  about  300  000.  kilometers  per  second. 

ANGULAR  VELOCITIES;  ROTARY  SPEEDS. 

(Angle  -r-  time.) 

For  simple  reductions  or  relations  between  angles,  see  values  under  Angles. 
Aprx.  means  within  2%. 
Angular  velocity  =  angle  moved  through  divided  by  time. 

in  degrees  per  second  =  angle  in  degrees  -r-  time  in  seconds. 

=  revolutions  per  second  X  360. 

"     in  revolutions  per  minute  =  revolutions  -5- time  in  minutes. 
' '     in  radians  per  second  =  2  n  X  revolutions  per  second. 

Logarithm 
1  revolution  per  hour  [rev/h  or  rph]: 

==0.1  degree  per  second 1 .000  0000 

1  radian  per  minute: 

=         9.549  30  revolutions  per  hour.    Aprx.  19/2 0-979  9714 

=       0.954  930  degree  per  second.    Aprx.  subtract  >^o 1-979  9714 

=  0.002  652  58  revolution  per  second.    Aprx.  %'-*- 1  000 3-423  6689 

1  degree  per  second: 

10.  revolutions  per  hour 1  000  0000 

1.047  20  radians  per  minute.    Aprx.  add  J^o  •  •   0-020  0287 

=  0.166  667  revolution  per  minute  or  Y& 1-221  8487 

=  Moo  or  0.002  777  78  rev  per  second.    Aprx.  %i  -*•  100 3-4436975 

1  revolution  per  minute  [rev/min  or  rpm]: 

6.  degrees  per  second 0-778  1513 

=     0.104  720  radian  per  second.    Aprx.  2j^ -7-100 1. 020  0287 

=  0.016  666  7  rev  per  second  or  Ko 2-221  8487 

1  radian  per  second  [aj]: 

=   57  295  8  degrees  per  second.    Aprx.  57 .  .    1-758  1226 

=  0.159  155  revolutions  per  second    Aprx.  1(Hoo 1-201  8201 

I  revolution  per  second  [rev/s  or  rps]: 

60.  revolutions  per  minute. 1-778  1513 

=  6.283  185  radians  per  second.     Aprx.^sXlO 0-7981799 

FREQUENCY;    PERIODICITY;    PERIOD;    ALTERNA- 
TIONS.    (l--time;   time.) 

Frequency  or  periodicity  is  the  number  of  recurrences  of  some  periodic 
or  wave  motion  during  a  given  time;  this  time  is  always  understood  to 
be  a  second  unless  otherwise  stated;  the  frequency  always  refers  to  the 
number  of  complete  waves.  The  "number  of  alternations"  however, 
refers  to  the  number  of  changes  of  the  direction  of  the  motion  or  to  the 
reversals,  and  therefore  refers  to  half  waves,  and  is  always  equal  to  double 
the  frequency,  if  the  time  is  the  same.  The  period  is  the  time  of  one  com- 
plete wave  or  oscillation  and  is  therefore  the  reciprocal  of  the  frequency. 
The  term  frequency  is  the  one  most  generally  used,  and  always  refers  to  a 
second;  the  term  number  of  alternations  is  unfortunately  preferred  by 
some  and  when  it  refers  to  electric  currents  the  time  is  usually  a  minute; 
the  term  period  is  used  comparatively  rarely  as  a  measure,  its  use  being 
generally  limited  to  scientific  discussions. 


FREQUENCY. — LINEAR    ACCELERATIONS.  87 

In  mathematical  discussions  of  electric  alternating-current  problems  the 
frequency  is  often  replaced  by  an  angular  velocity,  generally  represented 
by  (a  and  measured  in  radians  per  second  (see  under  Angular  Velocities 
above).  Then  to  =  2irnt  in  which  o>  is  in  radians  per  second  and  n  is  the  true 
frequency  in  cycles  per  second,  a  cycle  being  here  considered  the  same 
thing  as  a  complete  revolution. 

The  frequency  is  also  equal  to  the  velocity  of  propagation  divided  by 
the  wave  length.     The  wave  lengths  are  therefore  measured  in  units  of 
length,  but  when  the  velocity  for  a  class  of  waves  is  a  constant  (as  those 
of   light  or  the  electromagnetic  waves),  the  wave  lengths  may  also  be  in- 
dicated in  units  of  time,  in  which  case  a  wave  length  becomes  equal  to  the 
period  of  the  wave.     Wave  length  should  not  be  confounded  with  the 
amplitude,  which  latter  measures  the  intensity  of  the  wave  and  has  noth- 
ing to  do  with  the  frequency,  period,  or  wave  length. 
If  n  is  the  frequency  per  second  [</>],  then: 
the  period  in  seconds  =  l/n; 
the  number  of  alternations  per  minute  =  120n. 
If  n  is  the  number  of  alternations  per  minute,  then: 
the  frequency  per  second  =  n/120; 
the  period  in  seconds  =  120/n. 
If  n  is  the  period  in  seconds,  then: 

the  frequency  per  second  =  l/n; 
the  number  of  alternations  per  minute  =  120n. 

If  n  is  the  frequency  in  cycles  per  second,  a>  the  angular  velocity  in 
radians  per  second,  and  if  a  cycle  is  represented  by  one  revolution,  then: 
o)  =  2xn;  or 
n  =  0.159  155  <u: 
w  =  6. 283  19  n 
An  electrical  degree  is  the  360th  part  of  one  complete  cycle. 

LINEAR  ACCELERATIONS;    RATE    of  INCREASE  in 
VELOCITIES;  GRAVITY.     (Velocity -time.) 

The  mean  value  for  "gravity"  which  has  been  taken  as  a  basis  in  all 
these  tables  is  an  acceleration  of  9.805  966  meters  per  second  per  second, 
at  sea-level  and  in  latitude  45°.  This  is  probably  the  best  available  mean 
value  afid  is  known  as  Helmert's  value  (Die  math.  u.  phys.  Theorien  der 
hoehern  Geodaesie;  II,  p.  241;  1881).  It  is  used  up  to  date  by  the  Inter- 
national Geodetic  Association,  according  to  its  latest  published  report. 
No  value  has  been  adopted  by  the  National  Bureau  of  Standards.  At  the 
International  Bureau  of  Weights  and  Measures  the  value  9.809  91  is  taken, 
from  which  the  normal  value  at  sea-level  and  in  latitude  45°  would  be 
9.806  65;  the  difference  is  only  about  7  in  one  hundred  thousand. 

This  value  enters  into  many  figures  involving  relations  between  forces 
and  masses,  as,  for  instance,  in  the  relation  between  watts  and  horse-power, 
because  a  watt  is  defined  in  terms  of  a  dyne  (that  is,  a  true  force,  which  is 
independent  of  gravity),  while  a  horse-power  is  defined  in  terms  of  pounds 
or  kilograms  (that  is,  masses  which  must  be  multiplied  by  the  value  of 
gravity  to  be  reduced  to  true  forces  comparable  with  dynes).  The  value 
of  gravity,  however,  falls  out  in  all  figures  involving  the  relation  between 
two  masses  when  both  are  considered  as  forces,  as,  for  instance,  in  the  rela- 
tion between  foot-pounds  and  kilogram-meters,  which  involve  pounds  and 
kilograms  considered  as  forces.  It  does  not  enter  at  all  in  the  inter-rela- 
tions between  watts,  joules,  ergs,  and  dynes,  as  these  are  the  same  through- 
out the  universe.  Pounds,  kilograms,  etc.,  are  really  masses,  and  as  such 
are  the  same  throughout  the  universe,  but  when  considered  as  weights 
or  forces  their  values  depend  on  gravity  and  are  slightly  different  on  differ- 
ent parts  of  the  earth;  when  measured  on  a  beam  balance  by  comparison 
with  other  weights,  their  values  would  remain  the  same,  but  when  meas- 
ured on  a  spring  balance  their  values  would  change  slightly,  as  the  spring 
measures  the  force  of  gravity  acting  on  the  masses. 

The  "gravity"  here  referred  to,  which  is  an  acceleration  and  pertains 
only  to  our  earth,  should  not  be  confounded  with  what  is  called  the  "gravi- 
tation constant"  or  "Newton's  constant,"  which  pertains  to  the  whole 
universe,  and  is  the  constant  by  which  one  must  multiply  the  product  of 


88          LINEAR    AND    ANGULAR    ACCELERATIONS. 

the  masses  of  two  bodies  divided  by  the  square  of  their  distance  apart,  in 
order  to  get  the  force  of  attraction  between  them. 
Aprx.  means  within  2%. 

Logarithm 

1  kilometer  per  hour  per  min.  [km/hr/min](or  per  min.  per  hour): 
=  0.016  666  7  or  Ko  kilometer  per  hour  per  second  (or  per 

second  per  hour)  which  see  for  other  values.    2-221  8487 
I  mile  per  hour  per  minute  [ml/hr/min]  (or  per  minute  per  hour): 
=  0.016  666  7  or  %Q  mile  per  hour  per  second  (or  per  sec.  per 

hr.),  which  see  for  other  values 2-221  8487 

I  centimeter  per  second  per  second  [cm/s2]: 

0.036  kilometer  per  hour  per  second  (or  per  second 

per  hour) 2-556  3025 

=   0.032  808  3  foot  per  second  per  second.    Aprx.  V^-r-lO.  ..   2-5159842 
=   0.022  369  3  mile  per  hour  per  second  (or  per  second  per 

hour).    Aprx.  %  -r- 100 2-349  6529 

=  0.001  019  79  gravity.    Aprx.  Vi  ooo .' 3-008  5096 

1  kilometer  per  hour  per  second  [km/hr/s]  or  per  second  per  hour): 
=  27.777  8  centimeters  per  second  per  second.  Aprx.  28  1-443  6975 
=  0.911  343  foot  per  second  per  second.  Aprx.  subtr.  ^ij. .  1-959  6816 
=  0.621  370  mile  per  hour  per  second  (or  per  second  per 

hour).    Aprx.  %, 1-793  3503 

=     0.277  778  meter  per  second  per  second.    Aprx.  ^ii .....    1-443  6975 

=  0.028  327  4  gravity.    Aprx.  %  -*• 10 2-452  2071 

1  foot  per  second  per  second  [ft/s2]: 

—  30.480  1  centimeters  per  second  per  second.  Aprx.  30.  .   1.4840158 
=       1.097  28  kilometers  per  hour  per  second  (or  per  second 

per  hour).  Aprx.  add  ^o 0-040  3184 

=     0.681  818  mile  per  hour  per  second  (or  per  second  per 

hour).    Aprx.  4Mo 1-833  6687 

=     0.304  801  meter  per  second  per  second.    Aprx.  s/\o 1-484  0158 

=  0.031  083  2  gravity.    Aprx.  31  -r-1  000.  .  . 2-492  5254 

1  mile  per  hour  per  second  [ml/hr/sec]  or  per  second  per  hour: 

=       44.704  1  centimeters  per  sec  per  sec.    Aprx.  %  X  10. . .  .    1.650  3471 
=       1.609  35  kilometers  per  hour  per  second  (or  per  second 

per  hour).    Aprx.  % 0-206  6497 

=       1.466  67  feet  per  second  per  second.    Aprx.  4%o 0-166  3313 

=     0.447  041  meter  per  second  per  second.    Aprx.  %-i-lO.  .    1.650  3471 

=  0.045  588  6  gravity.    Aprx.  %-j-100 2-658  8567 

1  meter  per  second  per  second  [m/sec2]: 

3.6  kilometers  per  hr.  per  sec.  (or  per  sec.  per  hr.) . .  0-556  3025 

=    3.280  83  feet  per  second  per  second.    Aprx.  MX  10 0-515  9842 

=   2.236  93  miles  per  hr  per  sec  (or  per  s  per  hr).  Aprx.  %..   0-349  6529 

=  0.101  979  gravity.    Aprx.  Vio 1-008  5096 

Gravity  =  980.596  6  centimeters  per  sec  per  sec.  Aprx.  1  000. .    2-991  4904 
=       35.301  5  kilometers  per  hour  per  second  (or  per 

second  per  hour).  Aprx.  %  X  10 1.547  7929 

=  32.171  7  feet  per  second  per  second.  Aprx.  32.  ..  1-507  4746 
=  21.935  3  miles  per  hr  per  sec  (or /sec/hr).  Aprx.  22  1-3411433 
=  9.805  966  meters  per  second  per  second.  Aprx.  10.  0-991  4904 

ANGULAR   ACCELERATIONS;    RATE    of  INCREASE 
in  ANGULAR  VELOCITIES.     (Angular  velocity -H  time.) 

Logarithm 
V  revolution  per  minute  per  minute  [rev/min/min]: 

=      0.016  666  7  or  Ko  rev  per  min  per  sec  (or  per  s  per  m). .   2-2218487 

=  0.000  277  778  or  Meoo  rev  per  s  per  s.    Aprx.  3/n  -=- 1  000.  .    4.443  6975 
1  rerolution   per  minute  per  second  [rev/min/s]  or  per 

second  per  minute: 

60.  revolutions  per  minute  per  minute 1-778  1513 

=  0.016  666  7  or  Ko  revolution  per  second  per  second 2-221  8487 

1  revolution  per  second  per  second  [rev/s/sl: 

=  3  600.  revolutions  per  minute  per  minute. 3-556  3025 

—  60.  rev  per  minute  per  second  (or  per  sec  per  min.) 1-778  1513 


ANGLES. 


ANGLES   (plane)  ;   CIRCULAR  MEASURE. 

Aprx.  means  within  2%.  Logarithm 

1  second  ["1  =  0.016  6667  minute,  or  Vnn. 2.291  R4R7 

1  mill 

"  "  =         0.016  666  7    degree,  or  y60. ". ".' .  .  .  .  .  .  .  .  .  -  .  .  .  .  §221  8487 

=     0.000  290  888   radian 4-463  7261 

=     0.000  185  185   quadrant 4-267  6062 

=  0.000  046  296  3  circumference 5-665  5462 

1  grade  =0.01  right  angle 2-000  0000 

1  degree  \°~\: 

3  600.  seconds ! 3-556  3025 

60.  minutes 1-778  1513 

=   0.017  453  3   radian.    Aprx.  %  +  WQ 2-241  8774 

=   0.011  111  1    quadrant,  or  Voo 2-045  7575 

=  0.00555556    TT'S  (considered  as  an  angle  of  180°),  or  M.SO.  .  3-7447275 

=  0.002  777  78   circumference,  or  Mieo 3.443  6975 

1  radian: 

=  angle  whose  arc  equals  radius. 

=     206  265.  seconds .  5-314  4252 

=    3  437.75   minutes.     Aprx.  3400.  . 3.536  2739 

=    57.295  8   degrees.     Aprx.  57 1-758  1226 

=  0.636  620   quadrant.     Aprx.  Vn 1-803  8801 

=  0.318  310    TT'S  (considered  as  an  angle  of  180°).  Ap.  32/100 .  .  j.502  8501 

=  0.159  155   circiimference.     Aprx.  1(Kioo 1-201  8201 

1  quadrant  or  right  angle: 

=  100.  grades 2-000  0000 

=  90.  degrees 1-954  2425 

=  1.570  80  radians.     Aprx.  n/7 0-196  1199 

=  0.5  TT'S  (considered  as  an  angle  of  180°),  or  M 1-698  9700 

=         0.25  circumference,  or  '4 1-397  9400 

TC  (as  an  angle)  or  wcnu-circumlereiice: 

=           180.  degrees > 2-255  2725 

=  3.141  59   radians.     Aprx.  2Jft 0-497  1499 

2.  quadrants 0-301  0300 

=  0.5  circumference,  or  Yi 1-698  9700 

1  circumference  or  revolution  [rev] 

=       21  600.  minutes 4-334  4538 

=  360.  degrees 2-556  3025 

=  6.283  185  radians.     Aprx.  %X10 0-798  1799 

=  4.  quadrants 0-602  0600 

=  2.  TT'S  (considered  as  an  angle  of  180°) 0-301  0300 

1  electrical  degree  =the  360th  part  of  a  cycle;  see  p.  121. 

SOLID    ANGLES.     (Surfaces  radius.) 

A  solid  angle  is  an  angle,  like  that  at  the  point  of  a  cone,  which  is  sub- 
tended by  a  spherical  surface.  The  unit  solid  angle  is  that  angle  which, 
at  the  center  of  a  sphere  of  unit  radius,  subtends  a  unit  area  on  the  surface 
of  the  sphere;  this  unit  is  sometimes  called  a  steradian.  A  spherical 
right  angle  is  assumed  to  be  an  angle,  like  at  the  corner  of  a  rectangular 
block,  which  is  bounded  by  three  planes  perpendicular  to  each  other. 

Logarithm 

1  unit=     0.636  620  spherical  right  angle,  or  2A 1-803  8801 

"       =     0.159  155  hemisphere,  or  l-^27r 1-2018201 

"       =0.079577  5  sphere,  or  1+4* 2-900  7901 

1  steradian  =  1  unit  solid  angle ;  ( see  above) 0  000  0000 

1  spherical  right  angle  =1.570  80  units,  or  x/2 0-196  1199 

=         0.25  hemisphere,  or  M 1-397  9400 

=       0.125  sphere,  or  H 1-096  9100 

1  hemisphere  =  6-283  19  units,  or  2?r 0-798  1799 

=  4.  spherical  right  angles 0-602  0600 

=  0.5  sphere,  or  M 1-698  9700 

1  sphere  =  12.566  4  units,  or  4* 1-099  2099 

8.  spherical  right  angles 0-903  0900 

44          =  2.  hemispheres 0-301  0300 


90  GRADES;  SLOPES;  INCLINES. 


GRADES;  SLOPES;  INCLINES.     (Angle;  length -4- length.) 

Grades  are  indicated  in  terms  of  so  many  different  kinds  of  units,  some 
of  which  involve  trigonometric  relations,  that  the  relations  between  some  of 
them  become  complicated.  Those  given  in  the  following  table  are  mathe- 
matically correct  and  are  strictly  proportional;  they  may  therefore  be 
used  like  those  in  the  other  tables;  for  instance,  a  1%  grade  from  the 
table  is  52.8  feet  rise  per  mile,  hence  a  5%  grade  will  be  5  times  this;  or 
from  the  table,  1  foot  pter  mile  is  a  0.018  9%  grade,  hence  100  feet  per  mile 
will  be  a  1.89%  grade. 

Much  unnecessary  labor  and  confusion  would  be  avoided  if  grades  were 
uniformly  represented  in  percent,  that  is,  in  the  rise  per  hundred.  There 
seems  to  be  a  tendency  to  adopt  this  unit  generally.  In  the  following 
table  all  the  relations  are  given  in  terms  of  the  percent  unit  as  a  basis ;  the 
relation  between  any  two  others  is  readily  found  by  reducing  both  to  the 
same  percentage  value. 

In  using  percentage  values  it  should  be  remembered  that  they  mean 
the  rise  per  hundred,  and  it  may  therefore  be  necessary  sometimes  to 
multiply  or  divide  by  100  when  the  actual  or  total  distances  or  rises  are 
involved.  Thus  a  rise  of  12  feet  in  600  feet  is  a  2%  grade,  as  the  600  feet 
must  first  be  divided  by  1OO  to  reduce  it  to  hundreds,  before  dividing  it 
into  12.  Or  if  the  total  rise  on  a  2%  grade  is  given  as  12  feet,  it  means 


12-^2  =  6  feet  per  hundred,  which  must  therefore  be  multiplied  by  100 
!  total  dis 

irises  from  the  incorrect  way  in  which  perc         _ 
are  not  infrequently  written.     Thus  fifty  percent  should  be  written  50% 


to  get  the  total  distance. 

Much  confusion  arises  from  the  incorrect  way  in  which  percentage  values 


and  not  .50%,  which  latter  means  half  of   one  percent,  or  fifty  huridredths 
of  one  percent. 

Much  confusion  also  arises  from  the  fact  that  sometimes  the  sloping  or 
inclined  distance  is  meant  instead  of  the  horizontal  distance,  and  gener- 
ally it  is  not  stated  which  one  is  understood.  In  referring  to  profiles  or  to 
distances  on  a  map,  the  horizontal  distance  is  always  understood,  thus 
involving  the  tangent  of  the  angle;  while  in  formulas  for  the  traction  on 
grades,  or  when  the  distances  are  the  actual  lengths  of  track  or  road,  the 
sloping  or  inclined  distance  is  generally  irmplied,  as  the  traction  formulas 
then  become  simpler;  they  then  involve  the  sine  of  the  angle.  The  pres- 
ent table  gives  the  correct  reduction  factors  for  both.  Some  of  these  are 
the  same  in  both  systems;  but  it  should  be  understood  that  a  grade  of  a 
given  percent  based  on  horizontal  distances  is  slightly  different  from  a 
grade  of  the  same  percent  based  on  sloping  distances;  the  latter  is  always 
the  larger  angle  or  steeper  grade.  The  difference  is  however  generally 
negligibly  small  for  all  but  exceptionally  steep  grades;  up  to  a  14%  grade, 
which  is  about  the  limit  for  traction  on  rails  and  for  the  usual  roads,  the 
difference  is  less  than  1%. 

The  following  reduction  factors  are  the  same  whether  the  units  are 
based  on  the  horizontal  or  on  the  sloping  distances. 

Logarithm 

1  inch  per  mile  [in/ml]  =0.001  578  28% 3-198  1849 

1  foot  per  mile  [ft/ml]  =    0.018  939  4% 2-277  3661 

1  %o  0.1% 1.000  0000 

1  per  mil  [%0~\  0.1% 1. 000  0000 

1  millimeter  per  meter  [mm/m]  =  0.1% 1-000  0000 

1  foot  per  thousand  feet  [ft/M]    =  0.1% 1-000  0000 

1  foot  per  1OO  feet  [ft /C]  1.% 0-000  0000 

1  foot  rise  per  foot  [ft /ft]  100.% 2-000  0000 

1%  =633.6  inches  per  mile 2-801  8152 

"   =  52.8  feet  per  mile 1-722  6339 

"    =     10.  %9,  or  10 per  mil.; 1-0000000 

"   =     10.  millimeters  per  meter ' 1-000  0000 

"    =     10.  feet  per  thousand  feet 1-000  0000 

"    =       1.  foot  per  hundred  feet.. 0-000  0000 

"    =  0.01  foot  rise  per  foot 2-000  0000 


GRADES;  SLOPES;  INCLINES.  91 

n  miles  per  foot  rise  =  1/n  feet  rise  per  mile. 

=  (0.0189394  -s-n)%. 

n%  =(0.018  939  4  -*-n)  miles  per  foot  rise. 
n  feet  (rise)  per  mile  =  1/n  miles  per  foot  rise. 

n  feet  per  foot  rise   =  1/n  feet  (rise)  per  foot. 

=  (iop/n)%. 

n%  =  (I00/n)  feet  per  foot  rise. 

n  loot  (rise)  per  foot  =  1/n  foot  per  foot  rise. 

The  following  reduction  factors  are  only  for  units  based  on  the    hori- 
zontal distances. 

n%  =  —7:  —  %  based  on  sloping  distances. 


n%  =  100Xsin. 

n  degrees  rise  =  (100  tan  n)%. 

n%  =  number  of  degrees  whose  tangent  is  0.01  n. 

If  n  is  the  tangent  of  angle,  then  the  rise  in  %  =  100n. 
If  n  is  the  rise  in  %,  then  the  tan  =  O.Oln. 

If  n  is  the  sine  of  angle,  then  the  rise  in  %  =  —  Or  find  from  a 

vi  -n2 
table  the  tangent  corresponding  to  this  sine,  then  the  rise  in  %  =  100  X  tan. 

If  n  is  the  rise  in  %,then  the  sine  of  the   angle  =      ,  —  .      Or 

Vl002  +  n2 
the  sine  may  be  found  from  a  table  as  that  corresponding  to  tan  =  0.01n. 

The  following  reduction  factors  are  only  for  units  based  on  the  sloping 
distances. 

n%  =  —  %  based  on  horizontal  distances. 

V1002-n2 
n%  =  100Xtan. 

n  degrees  rise  =  (100  sin  n)%. 

n%  =  number  of  degrees  whose  sine  is  O.Oln. 

If  n  is  the  sine  of  angle,  then  the  rise  in  %  =  100  n. 
If  n  is  the  rise  in  %,  then  the  sine  =  0.01  n.  ' 

If  n  is  the  tangent  of  angle,  then  the  rise  in  %=    100n_.     Qr  find 


from  a  table  the  sine  corresponding  to  this  tangent,  then  the  rise  in  %  = 
100  X  sin. 

If  n  is  the  rise  in  %,  then  the  tangent  of  the  angle  =  —7^  l  Or 

Vl002-n2 
the  tangent  may  be  found  from  a  table  as  that  corresponding  to  sine  =  O.Oln. 


Approximate:  for  small  angles  and  for  most  engineering  calculations 
concerning  grades,  the  sine  and  the  tangent  are  very  nearly  equal,  hence 
the  simpler  of  the  formulas  given  above  can  in  most  cases  be  used  for  both, 
and  no  tables  are  then  necessary.  Up  to  a  14%  grade  (about  8  degrees) 
the  error  made  thereby  is  less  than  1%. 

The  actual  values  given  below  in  the  fifteen-column  table  avoid  the 
calculations  with  the  above  reduction  factors.  Intermediate  values  suffi- 
ciently accurate  for  most  purposes  may  be  found  from  this  table  by 
ordinary  interpolation. 


92 


GRADES;  SLOPES;  INCLINES. 


For  Sloping  Distances  Only. 
Sine  Functions. 

Equivalent  Percent 
Based  on  Horizontal 
Distances. 

OOi-HCOCO    r-H>COt>"3    i"~  Ci  i-<  "*<  l> 

§OOOO    i-H*-H<NCOO    OOi-Hr-li-l 

0000   0000^   ^^^^^ 

>-«MOO^lO    COr^XC5r-l    i-t  i-H  r-t  r-l  r-t 

1 
1 

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OOOOO   OOOOO   OOOOO 

a 

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.-KMCOTFiO    O1>XO;O   r-KNCO-^iO 

OOOOO   OOOOO   OOOOO 

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CO       rf^H»OC^       CO  r-4  -tf   1-1  «O  <N       CO 

o-H'-KNfN  eo^^ioio  co<£>r>.QOoo 

For  Horizontal  Distances  Only. 
Tangent  Functions. 

Equivalent  Percent 
Based  on  Sloping 
Distances. 

§o>O5OiC5  co  oo  i>  co  10  O5CJOOCOX 
O  O5  O5  Oi   C5  O  O5  Ol  O5       

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TIME.  93 


TIME. 

There  are  in  use  two  different  systems  of  units  of  time,  the  mean  solar 
time  and  the  sidereal  time.  The  mean  solar  time  is  based  on  the  appar- 
ent motion  of  the  sun  with  respect  to  the  earth,  that  is,  on  the  motion  of  the 
earth  with  respect  to  the  sun.  The  sidereal  time  is  based  on  the  appar- 
ent motion  of  the  stars  with  respect  to  the  earth,  that  is,  on  the  motion  of 
the  earth  with  respect  to  the  stars. 

The  mean  solar  time  is  that  indicated  by  the  clocks  in  common  use; 
it  is  that,  which  is  furnished  by  the  U.  S.  Naval  Observatory  and  is  used 
in  ail  physical  research.  In  all  derived  units  in  use,  such  as  velocities, 
forces,  power,  electrical  quantities,  etc.,  which  involve  the  element  of  time, 
this  mean  solar  time  is  understood  to  be  meant.  According  to  the  National 
Bureau  of  Standards,  Lord  Kelvin  (formerly  Sir  William  Thomson),  Prof. 
Walter  S.  Harshrnan  the  Director  of  the  Nautical  Almanac,  Prof.  R.  S. 
Woodward,  and  other  authorities,  it  is  the  second  of  mean  solar  time  which 
is  the  unit  of  time  in  tho  ccntimctcr-gram-second  system  of  units.  Mean 
solar  time  is  always  understood  to  be  meant  in  all  designations  of  time  un- 
less otherwise  specifically  stated.  In  all  the  units  in  this  book  which  involve 
time,  the  mean  solar  time  is  understood. 

Sidereal  time  is  used  only  for  astronomical  purposes.  It  is  considered 
as  possessing  more  nearly  the  essential  qualification  of  a  standard  unit, 
namely  invariability,  but  the  mean  solar  time  has  been  adopted  instead. 
However,  the  relation  between  the  two  is  known  to  such  a  degree  of  pre- 
cision, that  mean  solar  time  is  also  perfectly  uniform.  Tables  for  inter- 
changing sidereal  and  mean  solar  time  are  given  in  the  American  Ephemeris. 

Besides  these  two  sets  of  units  of  time,  there  are  probably  hundreds  of 
other  terms  used  to  designate  various  periods  of  time,  chiefly  different 
kinds  of  years,  months,  cycles,  etc.  These  are  generally  used  only  in 
astronomy  and  history,  and  many  of  them  are  obsolete;  they  have  there- 
fore not  been  included  in  tha  following  table. 

All  the  important  values  in  the  table  have  been  checked  through  the 
kindness  of  Prof.  Walter  S.  Harshman,  Director  of  the  Nautical  Almanac; 
many  of  these  have  been  accepted  as  the  best  obtainable  values,  and  aa 
such  are  used  in  the  American  Ephemeris  and  Nautical  Almanac. 

The  International  Bureau  of  Weights  and  Measures  has  established  an 
important  distinction  in  the  notation  of  time.  When  it  refers  to  the  epoch, 
that  is,  the  date  or  time  of  day,  the  reference  letters  are  used  as  indices; 
and  when  it  refers  to  the  duration  of  a  phenomenon,  they  are  on  the  same 
line  with  the  numbers.  For  instance,  an  experiment  began  at  2h  15m  46" 
lasted  2h  15m  46s,  and  ended  at  4h  31m  32s. 

Standard  Hail  way  Time  in  the  United  States  and  Canada.  On 
November  18,  1883,  a  new  system  of  railway  time  called  "Standard  Time  " 
went  into  effect  on  most  of  the  railroads  of  the  United  States  and  Canada, 
and  has  since  been  adopted  by  most  of  the  principal  cities.  According  to 
this  system  the  country  is  divided  into  five  strips  or  zones  running  north 
and  south,  each  15°  in  width.  Throughout  each  strip  the  time  of  the  clock 
is  the  same,  and  it  differs  from  that  in  the  two  neighboring  strips  by  pre- 
cisely one  hour;  for  instance,  when  it- is  4  o'clock  in  one  strip  it  is  5  o'clock 
in  the  next  one  east,  and  3  o'clock  in  the  next  one  west. 

The  following  table  gives  the  longitude  of  the  middle  line  of  each  strip; 
the  time  in  that  strip  is  the  correct  time  for  that  longitude.  The  actual 
lines  midway  between  these,  where  th?  time  changes  by  one  hour,  do  not 
always  correspond  exactly  with  the  theoretical  ones,  for  obvious  reasons. 
The  table  also  gives  the  name  by  which  that  time  is  designated  in  each 
strip  and  the  conventional  color  by  which  it  is  indicated  on  maps.  Eastern 
time  is  exactly  5  hours  later  than  Greenwich  time. 

Mefarom°G?len<w3icY.eSt  Name  of  Standard  Time'  Conventional  Color. 
60°                           Intercolonial  time  Brown 

75°  Eastern  time  Red 

90°  Central  time  Blue 

105°  Mountain  time  Green 

120°  Pacific  time  Yellow 


94  TIME. 


TIME. 

Mean  solar  time  is  the  time  in  universal  use  except  in  astronomy.     Un- 
less otherwise  stated,  mean  solar  time  is  understood  in  this  table. 
*  Accepted  by  Prof.  Walter  S.  Harshman.  Director  of  the  Nautical  Almanac. 

1  sidereal  second -0.997  269  57*  second  (mean  solar) I- 99^8 126* 

1  second  [s]  (mean  solar)  =  1.002  737  91*  sidereal  seconds 0-001  1874* 

1  sidereal  minute  =  60.*  sidereal  seconds. 

1  minute  [min  or  m]  (mean  solar)  =  60.*  seconds  (mean  solar). 

1  sidereal  hour  =  60.*  sidereal  minutes,  or  3  600.*  sidereal  seconds. 

1  hour  [h]  (mean  solar): 

=  60.*  minutes  (mean  solar),  or  3  600.*  seconds  (mean  solar). 
1  sidereal  day : 

=          86  164.1*   seconds  (mean  solar). 
=  86  400.*  sidereal  seconds. 

=  1  440.*  sidereal  minutes. 

=  24.*  sidereal  hours. 

23.  h,  56.  m,  4.091*  s  (mean  solar). 

=  0.997  269  57*   day  (mean  solar) I.ggg  8126* 

=  1  mean  solar  day  less  3  m  and  55;909*  s  (mean  solar). 
1  day  (mean  solar) : 

86  400.*  seconds  (mean  solar). 
=      86  636.555*  sidereal  seconds. 

1  440.*  minutes  (mean  solar). 
=  24.*  hours  (mean  solar). 

=  24.  sidereal  hours,  3m,  56.555*  s  sidereal  time. 

=  1.002  737  91*   sidereal  days 0-001  1874* 

=  1  sidereal  day  plus  3.  minutes  56.555*  seconds  sidereal. 
365.242  20*  mean  solar  days  =  366. 242  20*  sidereal  days. 
1  civil  or  calendar  day: 

=  1  day  (mean  solar) ;  is  reckoned  from  mean  midnight  to  mean  midnight. 
An  apparent  solar  or  a  natural  day  is  variable. 
An  astronomical  or  nautical  day  is  reckoned  from  mean  noon  to  mean 

noon. 

I  week  =  7.  days  (mean  solar). 
1  anomalistic  month ; 

=  27.  days,  13.  hours,  18.  minutes,  37.4  seconds  (mean  solar  ?). 
1  civil  or  calendar  month  [mo]: 

=  28,  29,  30,  and  31  days  (mean  solar). 
=  aprx.  Ma  year  (mean  solar). 
1  average  lunar  or  synodic  month: 

=  29.  days,  12.  hours,  44.  minutes,  2.8*  seconds  (mean  solar). 
=  29.530  59*   days  (mean  solar). 

1  average  sidereal  month  =  27.  days,  7.  hours,  43.  minutes,  11.5*  seconds. 
1  lunar  year  =  354.  days,  8.  hours,  48.  minutes,  34.  seconds  (mean  solar), 

=    12.  lunar  or  synodic  months. 
1  common  lunar  year  =  354.  days. 
1  year  [yr]  (mean  solar): 

=  366.242  20*  sidereal  days. 

=  365.242  20*  days  (mean  solar). 

365.  days,  5.  hours,  48.  minutes,  46.*  seconds  (mean  solar). 
=  1.  sidereal  year  less  20.  minutes,  26.9*  seconds  (sidereal). 

1  sidereal  year: 

=  366.256  399  2*  sidereal  days. 

=  365.256  360  4*  days  (mean  solar). 

=  365.  days,  6.  hours,  9.  minutes,  9.5*  seconds  (mean  solar), 

1.  mean  solar  year  plus  20.  rnin,  23.6*  sec  (mean  solar). 
1  civil  or  calendar  year,  ordinary  =  365.*  days  (mean  solar), 
leapt        =366.*  days  (mean  solar). 
1  ccmmon  year  =  1.  ordinary  civil  or  calendar  year. 
1  Julian  year  =  365.25*  days  (mean  solar). 

t  A  leap  year  is  one  whose  number  is  divisible  by  4,  except  when  the 
number  ends  in  two  ciphers,  then  it  must  be  divisible  by  400. 


DISCHARGES;  IRRIGATION.  95 

1  Gregorian  year  =  365.  days,  5.  hours,  49.  minutes,  12.  sec  (mean  solar). 

tropical  or  natural  year  =  l.  year  (mean  solar). 

anomalistic  year  =  365.  d,  6.  h,  13.  m,  53.*  s  (mean  solar). 
A  legal  year  is  obsolete. 

solar  cycle  =  28.*  Julian  years. 

century t=  100.*  civil  or  calendar  years. 

lunisolar  cycle  =  532.  years. 
1  milleuium  =  1  000.  calendar  years. 


DISCHARGES;    FLOW    of    WATER;    IRRIGATION 
UNITS  j    VOLUME  and  TIME.     (Volume  -*•  time.) 

(See  also  Volumes.) 

Discharges  (as  of  water)  are  generally  measured  in  terms  of  some  volume 
per  second  as  cubic  feet  per  second,  gallons  per  second,  cubic  meters  per 
second,  etc.  The  relations  between  them  are  therefore  the  same  as  between 
those  respective  volumes,  which  see  under  the  units  of  Volumes.  The 
same  is  true  if  they  are  given  per  minute.  When  one  is  per  second  and 
the  other  per  minute,  reduce  either  to  the  same  time  as  the  other  and  then 
use  the  table  of  volumes. 

The  only  unit  differing  from  these  is  the  miner's  inch  which  is  sometimes 
used  in  the  western  United  States  for  measuring  the  flow  of  water  in  streams, 
particularly  for  mining.  It  is  a  somewhat  vague  unit,  generally  insuffi- 
ciently denned,  and  has  so-called  "legal"  values  in  different  States,  which 
values  differ.  Its  value  varies  from  about  1.20  to  about  1.76  cubic  feet  per 
minute;  the  mean  is  generally  taken  as  about  1.5;  for  this  value  the  reduc- 
tion factors  are  given  in  the  following  table.  Aprx.  means  within  2%. 

Logarithm 

1  miner's  inch  =  1.5  cubic  feet  per  minute 0-176  0913 

=        0.187  013  gallon  per  second.    Aprx.  3/i6  ..  .   1-271  8717 
=  0.025  cubic  foot  per  second.  or*4o  ....   2-397  9400 

=  0.000707  925  cb.  meters  per  sec.     Ap.  ^ooo.  •  4-849  9875 

The  acre-foot  is  sometimes  used  as  a  unit  for  measuring  irrigation ;    it 
means  a  body  of  water  1  acre  in  area  and  1  foot  in  depth.     It  is  therefore 
really  a  true  unit  of  volume.     Its  chief  equivalents  are; 
1  acre-foot  =  325  851.  gallons. 

=     43  560.  cubic  feet. 
=  1  613.33  cubic  yards. 
"          «=  1  233.49  cubic  meters. 


t  The  twentieth  century  is  generally  assumed  to  have  begun  with  Janu- 
ary 1,1901. 


ELECTRIC  AND    MAGNETIC   UNITS. 


ELECTRIC  and  MAGNETIC  UNITS. 

General  Remarks.— In  the  following  tables  C.G.S.  refers  to  the  centi- 
meter-gram-second system  of  units;  elmg  means  electromagnetic;  elst 
means  electrostatic;  v  means  the  velocity  of  light  in  air,  which  is  here 
taken  as  equal  to  about  3X1010  centimeters  per  second,  which  value  has 
been  included  in  the  logarithms  of  those  relations  which  involve  this  v;  the 
word  "  about,"  used  with  such  derivatives  of  this  velocity,  means  that  they 
include  whatever  inaccuracy  there  is  in  this  velocity.  Aprx.  means  that 
the  simple  fractions  given  are  correct  within  2%.  The  values  of  the  derived 
figures  in  these  tables  are  generally  given  to  six  significant  figures  and 
seven-place  logarithms,  even  though  the  original  fundamental  data  may 
sometimes  not  warrant  such  accuracy;  the  object  is  to  enable  the  correc- 
tions due  to  any  subsequently  adopted  more  accurate  fundamental  values 
to  be  made  by  mere  proportion  instead  of  by  complete  recalculations. 

The  fundamental  values  of  the  electrical  units  used  in  these  tables 
are  those  adopted  by  the  International  Electrical  Congress  at  Chicago. 
Besides  the  exact  values  in  terms  of  the  C.G.S.  units,  that  Congress  also 
defined  certain  concrete  units  as  the  closest  approximations  which  existed 
at  that  time;  these  concrete  units  are  the  ones  in  use  in  practice  at  pres- 
ent. In  calculating  the  relations  between  the  electrical  and  the  mechan- 
ical and  thermal  units  like  foot-pounds,  horse-powers,  heat-units,  etc.,  for 
these  tables,  it  had  to  be  assumed  that  these  concrete  units  are  exactly 
equal  to  those  defined  in  terms  of  the  C.G.S.  units  which  they  represent,  as 
it  would  otherwise  be  impossible  to  calculate  those  relations  until  the  dif- 
ferences which  may  exist  between  the  concrete  and  the  exact  values  are 
known;  such  relations  therefore  must  always  involve  these  differences  if 
they  exist ;  whatever  they  may  be  they  are  absolutely  negligible  in  all  but 
the  most  refined  physical  research. 

The  absolute  values  of  the  electric  units  are  given  for  both  the 
usual  electromagnetic  system  and  the  less  usual  electrostatic  system,  but 
the  magnetic  units  have  been  confined  to  those  in  the  electromagnetic 
system,  as  the  others  are  rarely  if  ever  used;  the  dimensional  formulas  and 
interrelations  of  the  latter  are  given  in  the  table  of  Physical  Quantities 
in  the  Introduction.  In  a  paper  read  before  the  American  Institute  of 
Electrical  Engineers  in  July,  1903,  Dr.  A.  E.  Kennelly  suggests  the  prefixes 
ab-  or  abs-  to  the  names  volt,  ohm,  etc.,  to  designate  the  corresponding 
absolute  electromagnetic  units;  thus  abvolt,  absohm,  etc.,  mean  the 
absolute  or  C.G.S.  electromagnetic  units.  Similarly  the  prefix  abstat- 
designates  the  corresponding  absolute  electrostatic  units.  The  suggestion 
seems  a  good  one,  as  it  is  often  very  convenient  to  have  easily  remembered 
specific  names  for  the  absolute  electrical  units. 

The  American  Institute  of  Electrical  Engineers  has  adopted  the  rule  that 
vector  quantities  when  used  should  be  denoted  by  capital  italics.  This 
applies  chiefly  to  electromotive  force,  current,  and  impedance. 

Owing  to  the  numerous  important  and  very  useful  interrelations 
between  the  various  electric  and  magnetic  quantities,  many  of 
whbh  are  simple  unit  relations  like  Ohm's  law  or  Joule's  law,  there  have 
been  added  to  the  tables  of  these  units  numerous  formulas  giving  the  rela- 
tions between  quantities  measured  in  terms  of  various  different  electric  and 
magnetic  unite,  many  of  which  will  frequently  be  found  useful;  they  are 
correct  numerically  also,  and  may  be  used  like  any  other  formulas.  As 
there  is  a  very  large  number  of  such  relations,  only  those  likely  to  be  used 
are  given;  the  others  can  be  readily  derived  from  them.  Such  relations 
are  usually  given  in  the  form  of  algebraic  formulas,  but  the  method  adopted 
here  is  preferred  because  it  shows  directly  for  what  particular  units  the 
relations  are  numerically  correct.  Treatises  on  alternating  currents  should 
be  consulted  for  the  limiting  conditions  under  which  these  relations  apply 
to  alternating  or  other  periodically  varying  quantities. 

The  author  is  indebted  to  Prof.  W.  S.  Franklin  for  important  suggestions 
concerning  the  quantitative  interrelations  of  electric  and  magnetic  units 
when  their  intensities  are  varying  or  alternating.  Also  to  Dr.  Frank  A. 
Wolff,  Jr.,  Assistant  Physicist  of  the  National  Bureau  of  Standards,  for  his 
kindness  in  endorsing  the  correctness  of  a  number  of  the  more  important 
derived  values  of  the  electrical  units. 


RESISTANCE;  IMPEDANCE;  REACTANCE.         97 

Mean,  effective  and  maximum  values  in  periodically  varying; 
functions.  —  With  alternating  currents  the  instantaneous  values  of  both 
the  electromotive  force  and  the  current  vary  continually.  When  the 
arithmetical  mean  or  average  of  all  these  momentary  values  of  the  electro- 
motive force  or  current  is  taken,  it  is  called  the  mean  value;  this  value  is 
seldom  if  ever  used;  for  a  true  alternating  current  the  algebraic  mean  is 
always  zero  for  one  whole  period,  hence  the  arithmetic  mean  of  the  whole 
period  or  the  algebraic  or  arithmetic  mean  of  half  a  period  is  used  instead. 
When  the  square  root  of  the  mean  square  is  used  it  is  called  the  effective 
value;  this  is  the  value  almost  always  used  and  is  the  one  meant  when  not 
otherwise  specih'ed;  it  is  this  value  which  corresponds  to  the  electromotive 
force  or  current  of  a  direct  current  circuit  in  calculations  of  the  energy. 
Similarly,  there  is  a  mean  and  an  effective  magnetomotive  force,  mag- 
netizing force,  flux,  flux  density,  etc.,  but  as  the  arithmetic  mean  values 
of  these  are  of  less  importance  the  effective  values  are  the  ones  gen- 
erally understood  unless  otherwise  specified.  By  the  maximum  value 
of  any  of  these  is  meant  the  greatest  value  reached  in  one  period  ;  this  value 
is  sometimes  the  important  one,  notably  in  the  strain  on  the  insulation, 
which  depends  on  the  maximum  electromotive  force,  or  in  the  calculation 
of  the  hysteresis  loss,  which  depends  on  the  maximum  flux  density.  These 
terms,  mean,  effective,  and  maximum,  are  not  used  in  practice  in  connec- 
tion with  the  power  in  watts;  the  mean  watts  are  always  understood  and 
are  equal  to  the  product  of  the  effective  volts,  the  effective  amperes,  and 
the  cosine  of  the  angle  of  phase  difference. 

The  following  table  gives  the  relations  between  the  maximum,  effective, 
and  mean  values  when  the  variations  follow  the  sine  law.  By  mean  value 
is  here  meant  that  for  a  half  period,  as  the  algebraic  mean  for  a  whole  period 
is  always  zero. 

Aprx.  means  within  2%. 
Mean  value  (half  period).  Logarithm 

-effective  value  X  0.900  316  (or(2-^  ir)\/2).  Aprx.  subt.  10%..  1.954  3951 

=  maximum  va'ueX  0.636  620  (or  2/7:).    Aprx.f's  ............  1-803  8801 

Effective  value: 


mean  value  X  1.11072  (or  (w-J-4)\/2).    Aprx.  *%  ..........  0-045  6049 

maximum  value  X  0.707  107  (or  W2).    Aprx.  %o  .........   1-849  4850 

Maximum  value: 

=  mean  value  XI.  570  80  (or'  7T/2).     Aprx.  1^7  ...............  0-1961199 

=»  effective  value  X  1.414  21  (or  \/2).    Aprx.  l%  ............  0-150  5150 


RESISTANCE  [R,  r];  IMPEDANCE  [Z,  z]j  REACT- 
ANCE [X,  x].  (Electromotive  force  ~  current ;  lengthX 
resistivity  ~  cross-section.) 

These  units  are  used  to  measure  the  opposition  offered  to  the  passage  of 
a  current  through  a  circuit  or  part  of  a  circuit.  The  greater  the  resistan  e 
the  greater  this  opposition.  It  is  similar  to  the  mechanical  friction  of  a 
moving  body,  like  that  of  water  in  a  pipe,  although  the  analogy  does  not 
extend  to  the  numerical  laws,  nor  are  there  any  specific  units  for  mechan- 
ical frictional  resistance,  as  there  are  for  electrical  resistance.  According 
to  Ohm's  law  the  resistance  in  ohms  is  equal  to  the  electromotive  force  in 
volts  divided  by  the  current  in  amperes;  or  according  to  Joule's  law  it  is 
equal  in  ohms  to  the  power  in  wiitts  divided  by  the  square  of  the  current 
in  amperes;  or  to  the  square  of  the  voltage  divided  by  the  watts.  These 
relations  apply  to  direct  currents,  and  refer  to  the  true  or  "ohmic"  resist- 
ance of  the  conductor  itself,  which  is  dependent  only  on  the  material,  size, 
and  temperature  of  the  conductor.  They  also  apply  to  alternating  cur- 
rents when  there  is  no  reactance  in  circuit  (caused  by  inductance  or  capacity) 
in  which  case  the  effective  values  of  the  electromotive  force  and  current, 
and  the  true  watts,  are  meant.  Resistance  refers  to  a  given  circuit  or  part 
of  a  circuit,  while  resistivity  (which  see)  refers  to  the  specific  resistance  of  the 
material  irrespective  of  the  size  or  shape  of  the  circuit. 


98        RESISTANCE;  IMPEDANCE;  REACTANCE. 

Reactance,  although  not  a  resistance,  and  impedance,  which  is  often 
called  apparent  resistance,  and  is  a  resistance  combined  with  a  react- 
ance, are  both  correctly  measured  and  expressed  in  the  same  units  as  resist- 
ance, namely,  ohms,  although  the  reactance  depends  on  the  inductance  and 
capacity  of  the  circuit  and  on  the  frequency  of  the  alternating  current,  and 
is  therefore  not  true  resistance.  Both  of  these  terms  are  limited  chiefly  to 
alternating  current  circuits.  The  impedance  in  ohms  is  equal  to  the  effec- 
tive electromotive  force  in  volts,  divided  by  the  effective  current  in  amperes, 
regardless  of  what  the  phase  difference  may  be  (that  being  embraced  by  the 
vector  character  of  the  impedance).  It  is  also  equal  to  the  square  root  of 
the  sum  of  the  squares  of  the  resistance  and  the  reactance,  all  being  ex- 
pressed in  ohms.  For  further  explanations  concerning  the  calculations  of 
alternating  current  circuits,  reference  should  be  made  to  treatises  on  this 
subject. 

In  direct  current  circuits,  resistance  (true  or  "ohmic")  in  ohms  is  the 
reciprocal  of  conductance  in  mhos;  similarly,  in  alternating  current  cir- 
cuits impedance  in  ohms  is  the  reciprocal  of  admittance  in  mhos.  When 
various  parts  of  a  circuit  are  connected  in  series  their  total  resistance  or 
impedance  in  ohms  is  simply  the  sum  of  all  the  individual  resistances  or 
impedances  in  ohms.  If  they  are  in  parallel  or  multiple,  however,  it  is 
best  to  add  their  conductances  or  admittances  in  mhos  and  then  take  the 
reciprocal  of  this  sum,  which  will  then  be  the  total  joint  resistance  or  im- 
pedance in  ohms. 

The  unit  now  universally  used  is  the  ohm,  by  which  is  here  meant  the 
international  ohm  of  the  International  Congress  of  1893  at  Chicago,  based 
on  the  value  109  C.G.S.  units,  and  represented  by  the  resistance  of  a  column 
of  mercury  at  0°  C.,  106.3  centimeters  long,  weighing  14.452  1  grams,  and 
having  a  uniform  crpss-section.  It  was  made  legal  in  this  country  by  Con- 
gress in  1894  and  is  adopted  by  the  National  Bureau  of  Standards.  The 
Reichsanstalt  has  also  adopted  this  value,  and  there  is  therefore  uniformity 
in  the  resistance  standards  used  in  those  two  institutions.  The  National 
Bureau  of  Standards  at  present  uses  1-ohm  manganin  resistance  standards, 
verified  from  time  to  time  at  the  Reichsanstalt,  so  that  the  results  of  the 
Bureau  are  at  present  expressed  in  terms  of  the  particular  mercurial  resist- 
ance standards  of  the  Reichsanstalt.  The  construction  of  primary  mercurial 
standards  is  about  to  be  undertaken  by  the  Bureau.  The  British  National 
Physical  Laboratory  is  also  undertaking  the  construction  of  such  standards, 
and  the  present  definition  of  the  unit  of  resistance  in  that  country  in  terms  of 
the  Board  of  Trade  ohm  and  the  B.  A.  units,  may  be  replaced  by  one  in 
terms  of  the  primary  mercurial  standard. 

In  some  of  the  relations  with  other  units,  such  as  the  absolute  or  those 
of  energy,  the  value  of  the  international  ohm  as  above  defined  is  in  the 
following  tables  assumed  to  be  equal  to  the  theoretical  value,  namely, 
109  electromagnetic  C.G.S.  units,  which  value  is  sometimes  called  the  true 
ohm.  In  the  relations  between  the  international  ohm  and  the  other  mer- 
cury units  given  in  the  following  table,  it  is  assumed  that  the  uniform  cross- 
section  referred  to  in  the  definition  of  the  international  ohm  is  one  square 
millimeter. 

The  legal  ohm  was  a  mercury  standard  in  use  for  a  number  of  years 
prior  to  the  adoption  of  the  international  ohm ;  it  differs  from  the  latter  only 
m  that  the  length  is  106  centimeters  and  that  its  cross-section  is  defined  to 
be  1  sq.  millimeter,  while  with  the  international  ohm  it  is  the  weight  which 
is  defined. 

The  Siemens  unit  formerly  used  is  the  resistance  of  a  column  of  mer- 
cury, at  0°  C.,  having  a  cross-section  of  one  square  millimeter  and  a  length 
of  one  meter.  It  was  never  legalized,  but  has  often  been  used  as  a  well- 
defined  standard  of  reference. 

The  British  Association  unit,  or  B.  A.  unit,  was  formerly  the  standard 
in  Great  Britain ;  the  legal  standard  now  used  there  is  a  Board  of  Trade  coil 
or  ohm  equal  to  the  international  ohm,  on  the  assumed  relation  that  1  inter- 
national ohm  =  1.013  58  B.  A.  units.  The  relation  accepted  by  the  National 
Bureau  of  Standards  is  the  mean  of  the  relations  determined  by  Glazebrook 
and  by  Lindeck  in  1892,  namely,  1  international  (Reichsanstalt)  ohm  = 
1.013  48  B.  A.  units.  The  primary  standards  of  the  Reichsanstalt,  in  terms 
of  which  this  value  is  given,  are  themselves  subject  to  various  sources  of 
error  involved  in  the  construction  of  such  standards. 


RESISTANCE;  IMPEDANCE;  REACTANCE.         99 

The  electromagnetic  C.G.S.  unit  (or  absolute  unit)  is  the  resistance 
through  which  1  C.G.S.  unit  of  electromotive  force  will  cause  1  C.G.S.  unit 
of  current  to  flow. 

The  electrostatic  C.G.S.  unit  (or  absolute  unit)  is  similarly  denned 
with  respect  to  the  electrostatic  units  of  e.m.f.  and  current. 

RESISTANCE;   IMPEDANCE;   REACTANCE. 

**  Accepted  by  the  National  Bureau  of  Standards. 

*  Checked  by  Dr.  Frank  A.  Wolff,  Jr.,  Asst.  Phys.  National  Bureau  of 
Standards. 

Aprx.  means  within ,2%.  By  "ohm"  is  here  meant  the  international 
ohm,  unless  otherwise  stated,  v  is  the  velocity  of  light. 

Logarithm 

1  CGS  unit  [elmg]=         1  absohm 0-000  0000 

=  0.001  microhm 3-000  0000 

=  10~9  ohm 9-000  0000 

=   l/v2  CGS  unit  (elst).   About  %  X  lO"20..  .    21-0457575 

1  ahsohm  =  1  CGS  unit  (elmg) 0-000  0000 

1  microhm  =         1  000.  CGS  units  (elmg) 3-000  0000 

=  0.000  001   ohm 6-000  0000 

1  Siemens  unit  (S.U.)  =  0.940  734*  ohm.    Aprx.  subtract  6%.     1-973  4667 
1  Hi  it  ish  Association  unit  [B.A.U.]: 

=  0.986699**  ohm. f    Aprx.  subtract  1% 1-9941848 

=  0.986  602*    ohm.t    Aprx.  subtract  1% 1.994  1420 

I    eg-al  ohm  =  0.997  178*  ohm.    Aprx.  1 ' 1-998  7726 

1  ohm=  101J     CGS  units  (elmg)..  .• 9-000  0000 

10°     microhms 6. 000  0000 

"     =1.06300*    Siemens  units.    Aprx.    add  6% 0-0265333 

"     =1.01358*    British  Association  units. t    Aprx.  add  1%.     0-0058580 

"     =1.01348**  British  Ass'n  units. t    Aprx.  add  1% 0-0058152 

1     =  1.002  83*    legal  ohms.    Aprx.  1 0-001  2274 

' '     =          10~°     megohm 6-000  0000 

« •     =       I09/v2    CGS  unit  (elst).    About  Vo  X  10"11 12-045  7575 

1  international  ohm  =  1  ohm,  which  see  above. 

1  true  ohm  =  109  CGS  units  (elmg) 9-000  0000 

1  ohm  of  Reichsanstalt  =  1  ohm 0-000  0000 

1  Hoard  of  Trade  (Brit.)  ohiurT 

=  1.013  58  Brit.  Ass'n  unit.    Aprx.  add  1% 0-0058580 

1  ohm 0-000  0000 

1  megohm  =         10°  ohms 6-000  0000 

.  —  1015/v2  CGS  units  (elst).    About  %  X  10~5 6-045  7575 

1  CGS  unit  (elst)  =  v2  CGS  units  (elmg)  About  9X  1020  .    20-954  2425 

=  7>2X10-9ohms.    About  9X1011 11-9542425 

1  abstatohm 0-000  0000 

1  ahstatohm  =  1  CGS  unit  (elst) 0-000  0000 

1  absolute  unit  =  1  CGS  unit  either  elst  or  elmg 0-000  0000 

The  relations  to  other  measures  are  as  follows:  § 
Ohms  =  volts  -r- amperes. 

"      =  volts  X  seconds  -r-  coulombs. 
=  volts2  -T-  watts. 
=  watts  -r-  amperes2. 
=  watts  X  seconds2  -r-  coulombs2. 
=  volts2  X  seconds  -=-  joules. 
=  joules  -r-  (amperes2  X  seconds). 
=  joules  X  seconds  -J-  coulombs2. 

t  Mean  of  Glazebrook's  and  Lindeck's  values  of  B.  A.  units  in  terms  of 
Reichsanstalt  primary  mercurial  standards,  accepted  by  the  National 
Bureau  of  Standards. 

J  Legal  relation  in  Great  Britain. 

§  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  period- 
ically varying  quantities. 

1  Latest  value,  1903:  1  Reichsanstalt  ohm  =  1.000  165  Board  of  Trade  ohms. 


100      RESISTANCE;  IMPEDANCE;  REACTANCE. 

Ohms  resistance: 

=  1  -4- mhos  conductance.     For  direct  currents  only. 
=  henrys -Mime  constant  in  seconds. 

=  induced  volts -r-  (rate  of  change  of  amperes  per  second  X  time  con- 
stant in  seconds). 
=  henrys  X  final  amperes  X  applied  volts -^(  joules  of  kinetic  energy  of 

the  current  X  2). 
=  applied   volts  X\/[henrys-r-(  joules  of  kinetic  energy  of   the   current 

X2)]. 
=  applied  volts2  X  time  constant  in  seconds  -=-( joules  of  kinetic  energy  of 

the  current  X  2). 
=  joules  of  kinetic  energy  of  the  current  X  2  -*•  (time  constant  in  seconds 

X  final  amperes2). 
=  (maxwells  X  number    of    turns)  -r-  (final    amperes  X  time    constant    in 

seconds  X  10  ).     When  the  flux  is  due  only  to  the  current,  as  in 

self-induction. 

Microhms  resistance: 

-=l-hmegamhos  conductance.     For  direct  currents  only. 

For  alternating  current  circuits  :f 

Ohms  resistance : 

=  volts  energy  component  of  e.m.f. -=- total  amperes. 

=  watts  -r-  amperes2. 

=  V(ohms  impedance2  — ohms  reactance2). 

=  \/[(l  -r-mhos  admittance2)  — ohms  reactance2]. 

=  mhos  conductance  X  ohms  impedance2. 

=  mhos  conductance  -j- mhos  admittance2. 

=*\/(aPphed  volts2  — induced  volts2)-;- amperes. 

=  applied  volts -s-  (amperes  X  VlXtime  constant  in  seconds  X  frequency  X 

6.283  19t)2  +  l  ) 

=  V[(applied  volts •*- amperes)2- (henrys XfrequencyX 6. 283  19t)2]. 
=  vTohms  impedance2  —  (henrys  X  frequency  X  6.283  19$ )2]. 
^induced   volts -*- (time   constant   in   seconds X amperes X frequency X 

6.283  191:). 

Ohms  reactance  : 

=  volts  wattless  component  of  e.m.f. -r- total  amperes. 

=  \/(ohms  impedance2  — ohms  resistance2). 

=  VT(1  -s-mhos  admittance2)  —  ohms  resistance2]. 

=  ohms  impedance2  X  mhos  susceptance. 

=  mhos  susceptance^- mhos  admittance2. 

=  ohms  magnetic  reactance  —  ohms  capacity  reactance. 

=  (  henrys  XfrequencyX  6. 283  19J)- [0.159  155§  -^-(farads  X  frequency)]. 

Ohms  magnetic  reactance  =  henrys  XfrequencyX  6. 283  194 
Ohms  capacity  reactance  =  —0.159  155§  -K farads  X  frequency). 

=     —159  155. -r- (microfarads  X  frequency). 

Ohms  impedance  =  total  effective  volts  -*- total  effective  amperes, 
^^(ohms  resistance2-}- ohms  reactance2). 
=  1  -r-mhos  admittance. 

=  l-*-\/(inhos  conductance2  +  mhos  susceptance2). 
=  \/(ohms  resistance -T- mhos  conductance). 
=  v/(ohms  reactance -T- mhos  susceptance). 
=  \/[ohms  resistance2  -f  (henrys  X  freq.  X  6.283  19];  )2]. 

t  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  period- 
ically varying  quantities. 

t  Or  2*.    Aprx.  %  x  10.     Log  0-798  1799- 

§  Or  1-^2*.    Aprx.  %-*-10.     Log  1.201  8201- 


RESISTANCE    AND    LENGTH.  101 


RESISTANCE     and   LENGTH,    for    the,  SAME    CROSS- 
SECTION.     (Redstance-f-lcng'th.)  , 

For  wires  of  the  same  cross-section  and  material  the  following  relations 
exist.     Aprx.  means  within  2%. 

Logarithm 
1  ohm  per  mile: 

=        0.621  370  ohm  per  kilometer.     Aprx.  ^ 1-7933503 

=  0.000  621  370  ohm  per  meter.    Aprx.  Vs  •*•  1  000.. .  .  4.793  3503 

=  0.000  189  394  ohm  per  foot.    Aprx.  19-^-100  000  .  .  4.277  3661 
1  ohm  per  kilometer  : 

1.609  35  ohms  per  mile.     Aprx.  add  6/io 0-206  6497 

0.001  ohm  per  meter 3-000  0000 

=  0.000  304  801  ohm  per  foot.  Aprx.  3  -5-10  000 4-484  0158 

1  ohm  per  meter  =    1609.35  ohms  per  mile.    Aprx.  1  600 ....  3-2066497 

=         1  000.  ohms  per  kilometer 3-000  0000 

=  0.304  801   ohm  per  foot.    Aprx.  «/io 1-484  0158 

1  ohm  per  foot  =       5  280.  ohms  per  mile.    Aprx.  5  300 3-722  6339 

-=3280.83  ohms  per  klm.    Aprx.  M  X  10  000  .  .  3-5159842 

=  3.280  83  ohms  per  meter.    Aprx.  10/a 0-515  9842 


RESISTANCE     and     CROSS-SECTION,     for    the     SAME 
LENGTH.      (ResistanceX  cross-section.) 

When  the  lengths  of  wires  of  the  same  material  are  the  same  and  the 
cross-sections  are  different,  the  relations  between  the  respective  compound 
units  representing  the  product  of  the  resistance  and  the  cross-section  are 
the  same  as  the  relations  between  the  different  cross-section  units,  which 
see  under  the  units  of  Surface.  Thus  if  the  compound  unit  ohm-centi- 
meters2 is  the  product  of  the  resistance  in  ohms  and  the  cross-section  in 
square  centimeters  of  any  wire,  and  if  similarly  ohm-inches2  is  the  product 
of  the  oliJis  and  the  square  inches  cross-section  of  another  wire  of  the  same 
length,  then  1  ohm-centimeter2  =  0.1 55  000  ohm-inch2,  in  which  0.155000 
is  the  value  of  1  sq.  centimeter  in  sq.  inches. 

These  units  are  used  for  converting  the  values  of  resistivities  from  one 
unit  to  another  (see  also  under  units  of  Resistivity,  below);  thus  if  n  is  the 
resistivity  of  a  material  in  ohm,  circular  mil,  foot,  units,,  and  it  is  required 
to  change  this  into  ohm,  sq.  mil,  foot,  units,  multiply  'n  by  the  value  of 
1  circular  mil  in  square  mils  as  given  in  the  table  of  units  of  Surface,  namely, 
0.785  398. 


RESISTIVITY  [p]\    SPECIFIC  RESISTANCE. 

(Resistance  X  cross-section  ~-  length.) 

These  units  are  used  to  measure  the  inherent  quality  of  a  material  to 
resist  the  passage  of  an  electric  current.  Resistivity  differs  from  resistance 
in  that  the  latter  refers  to  the  number  of  ohms  of  any  given  circuit  or  part 
of  a  circuit,  and  depends  on  the  length,  cross-section,  and  quality  of  the 
material;  while  resistivity  refers  only  to  the  m-fi^  of  the  material  itself 
and  is  always  the  same  for  the  same  material;  it  is  the  resistance  of  a  unit 
amount  of  the  material,  like  a  cube  of  one  centimeter,  or  a  mil-foot,  or  a 
meter  of  one  sq.  millimeter  section.  It  bears  somewhat  the  same  relation 
to  resistance  as  the  density  of  a  material  does  to  the  weight  of  any  given 


102        RESISTIVITY;  SPECIFIC  RESISTANCE. 

amoiAit  of  it;  the  density  is  always  the  same  for  that  material,  being  the 
weight  of  a  unit  of  volume  .ifrhiie  the  total  weight  of  an  actual  piece  depends 
upon  the  size  of  that  piece.  *  As  the  resistivity  or  specific  resistance  is  a 
quality  of  a  material,  i^s,  values«are  usually  given  in  tables  of  physical  con- 
st put s^  the  refistivitv  may  also  be  calculated  from  the  resistance  of  any 
gi  /on  pi^.-e  by  multiplying  this  resistance  in  ohms  by  the  cross-section  and 
dividing  by  the  length;  the  result  will  of  course  be  different,  depending 
upon  the  units  used.  The  resistivity,  being  a  property  of  a  material,  is 
the  same  for  direct  as  for  alternating  currents. 

There  are  several  units  in  use.  The  most  rational  one  is  the  resistance 
in  ohms  between  two  parallel  sides  of  a  cube  of  one  centimeter  of  the  ma- 
terial. This  unit  is  sometimes  called  the  ohm-centimeter  unit,  or  1  ohm 
per  cubic  centimeter,  or  more  correctly,  1  ohm-square-centimeter 
per  centimeter;  it  will  here  be  called  the  ohm,  cubic  centimeter, 
unit.  For  most  conducting  materials  in  use,  except  electrolytes,  the 
resistivity  when  stated  in  these  units  is  a  very  small  number,  hence  it  is 
often  stated  in  microhms  (millionths  of  an  ohm)  instead  of  ohms. 

The  electromagnetic  and  the  electrostatic  C.G.S.  units  are  the 
same  as  the  above  except  that  the  resistance  is  stated  in  the  respective 
C.G.S.  units  instead  of  in  ohms. 

Another  unit  in  common  use  is  the  meter-millimeter  unit,  that  is,  the 
resistance  in  ohms  of  a  wire  one  meter  long  and  one  square  millimeter  in 
cross-section;  it  also  has  the  name  1  ohm  per  meter  per  square  milli- 
meter, or  more  correctly  1  ohm-square-millimeter  per  meter;  it  will 
here  be  called  the  ohm,  square  millimeter,  meter,  unit.  This  unit  has 
the  advantage  that  it  generally  involves  less  calculation  than  the  cubic- 
centimeter  unit,  as  the  lengths  and  cross-sections  are  in  practice  usually 
stated  in  meters  and  square  millimeters  when  the  metric  system  is  used.  A 
more  rational  unit  for  circular  wires,  although  apparently  not  in  general 
use,  is  similar  to  this  one  except  that  the  cross-section  is  a  circle  of  one 
millimeter  diameter,  and  is  therefore  equal  to  a  "circular  millimeter"  as 
distinguished  from  a  "square  millimeter"  (see  explanation  of  circular  units 
under  units  of  Surface).  This  has  the  advantage  of  eliminating  from  the 
calculation  for  the  usual  round  wires  the  troublesome  factor  n  (or  3.141  59), 
because  when  the  cross-sections  are  stated  in  circular  units  they  are  directly 
equal  to  the  squares  of  the  diameters. 

The  unit  usually  used  when  the  lengths  are  in  feet  and  the  diameters  in 
mils  (that  is,  thousandths  of  an  inch),  is  the  mil-foot  or  circular  mil-foot, 
which  is  the  resistance  in  ohms  of  a  round  wire  one  foot  long  and  one  mil 
in  diameter.  The  cross-section  then  is  one  circular  mil,  and  the  cross- 
sections  of  other  round  wires  are  then  equal  to  the  squares  of  their  diam- 
eters in  mils  (see  explanation  in  preceding  paragraph).  This  unit  also  has 
the  name  of  1  ohm  per  foot  per  circular  mil  pr  per  mil  diameter,  or  more 
correctly,  1  ohm-circular-mil  per  foot;  it  will  here  be  called  the  ohm, 
circular  mil,  foot,  unit.  A  similar  unit,  though  less  frequently  used,  is 
the  square  mil-foot;  it  differs  from  the  other  only  in  that  the  cross-section 
is  a  square  mil  instead  of  a  circular  mil,  and  it  is  more  convenient  to  use 
with  bars  of  rectangular  cross-section.  It  also  has  the  name  of  1  ohm  per 
foot  per  square  mil,  or  more  correctly,  1  ohm-square-mil  per  foot; 
it  will  here  be  called  the  ohm,  square  mil,  foot,  unit. 

Sometimes  resistivities  are  denoted  in  terms  of  that  of  mercury  or  pure 
copper  as  a  basis.  They  are  then  mere  relative  resistivities  or  ratios  of  two 
resistivities,  that  of  the  material  divided  by  that  of  mercury  or  copper,  and 
are  therefore  not  in  terms  of  any  real  units,  although  they  might  be  called 
mercury  or  copper  units.  In  the  following  table  the  resistivities  of  mercury 
and  of  copper  have  been  added  to  facilitate  making  calculations  with  such 
relative  resistivities.  The  resistivity  of  mercury  here  used  is  that 
deduced  from  the  definition  of  the  concrete  international  ohm,  namely,  that 
a  column  of  uniform  cross-section,  106.3  centimeters  long,  weighing  14.4521 
grams,  has  a  resistance  of  one  ohm  at  0°  C.  The  cross-section  is  for  this 
purpose  assumed  to  be  one  square  millimeter.  For  the  resistivity  of  pure 
copper  the  Matthiessen  value  is  still  in  use,  namely  1.687  microhms  for 
one  centimeter  length  and  one  square  centimeter  cross-section,  at  15°  C., 
according  to  Prof.  Lindeck  of  the  Reichsanstalt.  Pure  copper  as  now 
made  has  a  lower  resistivity  than  this;  according  to  Prof,  Lindeck  the 
value  used  in  Germany  (presumably  under  the  authority  of  the  Reichsan- 
stalt) is  1.667  in  the  same  units. 


RESISTIVITY;  SPECIFIC  RESISTANCE.         103 

Aprx.  means  within  2%.     v  is  the  velocity  of  light. 

Logarithm 

1  CGS  unit  (elmg)  =  10~~9  ohm,  cubic  centimeter,  unit 9-000  0000 

=  1/v2  CGS  unit  (elst).    About  %  X  lO"20. .  .    21.045  7575 
1  ohm,  circular  mil,  foot,  unit: 

=       0.785  398  ohm,  sq.  mil,  foot,  unit.    Aprx.  8/io 1-895  0899 

=       0.166  243  microhm,  cb.  cm,  unit.    Aprx.  Y& 1-220  7433 

=  0.002  116  67  ohm,  circ.  mm,  meter,  unit.    Aprx.  2Vio  ooo-     3-3256534 
=  0.001  662  43  ohm,  sq.  mm,  meter,  unit.    Aprx.  K  •*•  100..      3-2207433 
1  mil-foot  unit.     See  1  ohm,  circular  mil,  foot,  unit. 
1  ohm-circular-mil  per  foot.      See  1  ohm,  circular  mil,  foot,  unit. 
1  ohm  per  foot  per  circular.mil.     See  1  ohm,  circular  mil,  foot,  unit. 
1  ohm  per  foot  per  mil  diameter.     See  1  ohm,  circular  mil,  foot,  unit. 

1  ohm,  sq.  mil,  foot,  unit: 

=  1.273  24  ohm,  circular  mil,  foot,  units.  Aprx.  J%  0-1049101 
=  0.211  667  microhm,  cb.  cm,  unit.  Aprx.  21-^100..  .  L325  6534 
=  0.00269503  ohm,  circ.  mm,  met.  unit.  Aprx.  %  -*- 1  000.  3-4305635 
=  0.002  11667  ohm,  sq.  mm, met.  unit.  Aprx.  21  -4-10  000.  3-3256534 

1  ohm-sq.  mil  per  foot.      See  1  ohm,  sq.  mil,  foot,  unit. 

]  ohm  per  foot  per  sq.  mil.      See  1  ohm,  sq.  mil,  foot,  unit. 

1  microhm,  cb.  centimeter,  unit: 

1  000.  CGS  units  (elmg). .  . 
6.01529   ohm,  circular  mil,  foot,  units.    Aprx.  6.^.  .      0-7792567 

1/01 


1  000.  CGS  units  (elmg) 3-000  0000 

.01529   ohm,  circular  mil,  foot,  units.    Aprx.  6....      0-7792567 
=       4.72440   ohm,  sq.  mil,  foot,  units.    Aprx.  Hi  X  100.  .      0-6743466 


=  0.012732  4   ohm,  circ.  mm,  met.  unit.     Aprx.  >s -5-10...  .      2-1049101 

0.01    ohm,  sq.  mm,  meter,  unit 2-0000000 

10~6  ohm,  cb.  cm,  unit 6-000  0000 

1  microhm-sq.   centimeter  per  centimeter.       See  1  microhm,  cb.  cm, 

unit. 
1  microhm  per  cubic  centimeter.     See  1  microhm,  cb.  cm,  unit. 

1  ohm,  circular  mm,  meter,  unit: 

=                472.440  ohm,  circ.  mil,  ft,  units.  Aprx.  1°  °oo/21.  2-674  3466 

=                371.054  ohm,  sq.  mil,  ft,  units.    Aprx.  ^  X  1  000  2-569  4365 

78.539  8  microhm,  cb.  cm,  units.    Aprx.  80 1-8950899 

=            0.785  398  ohm,  sq.  mm,  meter,  unit.      Aprx.  8/io  :  -  1-895  0899 

=  0.000  078  539  8  ohm,  cb.  cm,  unit.    Aprx.  80-4- 1  000  000.  5-895  0899 

1  ohm-circular-mm  per  meter.      See  1'ohm,  circular  mm,  meter,  unit. 

1  ohm  per  meter  per  circular  mm.     See  1  ohm,  circular  mm,  meter, 

unit. 

1  ohm,  sq.  mm,  meter,  unit: 

=  601.529  ohm,  circular  mil,  foot,  units.  Aprx.  600.  .  .  .  2-779  2567 

=  472.440  ohm,  sq.  mil,  foot,  units.  Aprx.  j^i  X  10  000. .  2-674  3466 

=  100.  microhm,  cb.  cm,  units .  2-000  0000 

=  1.273  24  ohm,  circular  mm,  meter,  units.  Aprx.  1%  .  ..  0-104  9101 

=  0.000  1  ohm,  cb.  centimeter,  unit ;  4-000  0000 

1  ohm-sq.  mm  per  meter.     See  1  ohm,  sq.  mm, meter,  unit. 

1  ohm  per  meter  per  sq.  mm.     See  1  ohm,  sq.  mm,  meter,  unit. 

1  ohm.  cb.  centimeter,  unit: 

10«  CGS  units  (elmg) 9-000  0000 

=  1  000  000.  microhms,  cb.  cm,  units 6-000  0000 

=  12732.4  ohm,  circ.  mm,  met.  units.  Aprx.  i/s  X  100  000.  4-1049101 

=  10  000.  ohm,  sq.  mm,  meter,  units 4-000  0000 

=  109A2  CGS  unit  (elst).  About  VQ  X  10~ n 12-0457575 

1  ohm-sq.  cm  per  cm.     See  1  ohm,  cb,  centimeter,  unit. 

1  ohm  per  cubic  centimeter.     See  1  ohm,  cb.  cm,  unit. 

1  megohm,  cb.  centimeter,  unit: 

=  1  000  000.  ohm,  cb.  cm,  units 6-000  0000 

1  CGS  unit  (elst)  =  v2  CGS  units  (elmg).   About  9  X  1020  20-954  2425 

=  v2X10-9  ohm,  cb.  cm, units.    Abt.  9X1011  11-9542425 
"  =v2X  10~15  megohm,  cb.  cm,  units.      About 

9X  105 5-954  2425 


104  RESISTIVITY;   CONDUCTANCE. 


Logarithm 
Resistivity  of  copper:  f 

10.027  5    ohm,  circular  mil,  foot,  units 1-001  1923 

7.875  57    ohm,  sq.  mil,  foot,  units 0-896  2822 

1.667f  microhm,  cb.  cm,  units 0-221  9356 

=      0.021  224  9    ohm,  circular  mm,  meter,  unit 2-326  8457 

=      0.017  720  2    times  that  of  mercury 2-248  4689 

=  0.016  67    ohm,  sq.  mm,  meter,  unit 2-221  9356 

=  0.000001667    ohm,  cb.  cm,  unit 6-2219358 

Resistivity  of  copper  (Matthiessen):  J 

10.147  8    ohm,  circular  mil,  foot,  units 1-006  3718 

7.970  06    ohm,  sq.  mil,  foot,  units 0-901  4617 

1.687J  microhm,  cb.  cm,  units 0-227  1151 

=      0.021  479  5    ohm,  circular  mm,  meter,  unit 2-332  0252 

=      0.017  932  8    times  that  of  mercury 2-253  6484 

=  0.016  87    ohm,  sq.  mm,  meter,  unit 2-227  1151 

=  0.000  001  687    ohm,  cb.  cm,  unit 6-227  1151 

Resistivity  of  mercury  :  § 

565.879    ohm,  circular  mil,  foot,  units 2-752  7234 

444.40    ohm,  sq.  mil,  foot,  units 2-647  8133 

94.073  4    microhm,  cb.  cm,  units 1-973  4667 

56.432  7    times  that  of  copper 1.751  5311 

55.7637    times  that  of  copper  (Matthiessen).  .  .  .  1-746  3516 

1.197  78    ohm,  circular  mm,  meter,  units 0-078  3768 

=         O.94O  734§  ohm,  sq.  mm,  meter,  unit 1-973  4667 

=  0.000  094  073  4    ohm,  cb.  cm,  unit 5-973  4667 

The  relations  of  resistivity  to  other  measures  are  as  follows: 
Resistivity  (in  ohm.cb.  cm,  units)  =  Is- conductivity  (in  mho,  cb.  cm  units). 

=  ohmsXsq.  cm  section  •*•  cm  length. 


CONDUCTANCE  [G,  g];  ADMITTANCE  [Y,  y]j  SUS- 
CEPTANCE  [B,  b].  (Current  -=-  electromotive  force ; 
1  -T-  resistance ;  cross-section  X  conductivity  -*-  length.) 


These  units  are  used  to  measure  the  quality  of 


tne  resistances  are  generally  used  by  preference,  wnen,  However,  tnere 
are  several  circuits  in  parallel  or  multiple  arc,  the  calculation  of  their  joint 
action  is  simpler  if  made  with  conductances,  as  their  joint  conductance 
is  then  merely  the  sum  of  the  individual  conductances,  while  when  resist- 
ances are  used  the  joint  resistance  is  equal  to  the  reciprocal  of  the  sum  of 
the  reciprocals  of  the  individual  resistances. 

For  direct  current  circuits,  or  when  there  is  no  reactance  in  alternating 
current  circuits,  the  conductance  in  mhos  is  equal  to  the  reciprocal  of 
the  resistance  in  ohms.  It  follows  from  Ohm's  law  that  for  direct  current 
circuits,  and  for  alternating  current  circuits  without  reactance,  the  con- 
ductance in  mhos  is  equal  to  the  current  in  amperes  divided  by  the  elec- 
tromotive force  in  volts.  The  conductance  in  mhos  is  also  equal  to  the 
resistance  in  ohms  divided  by  the  sum  of  the  squares  of  the  resistance  in 
ohms  and  the  reactance  in  ohms.  The  relations  to  joules  and  watts  are 
rarely  if  ever  used. 

t  Pure  copper  at  15°  C.;  according  to  Prof.  Lindeck. 

j  Matthiessen's  value  for  pure  copper  at  15°  C. ;  according  to  Prof.  Lindeck. 

§  Pure  mercury  at  0°  C.,  based  on  definition  of  international  ohm. 


CONDUCTANCE;  ADMITTANCE;  SUSCEPTANCE.  105 

Admittance,  which  is  the  reciprocal  of  impedance,  and  susceptance, 
which  together  with  conductance  make  admittance,  are  both  correctly 
expressed  and  measured  in  the  same  units  as  conductances,  namely,  mhos, 
although  they  depend  on  the  inductance  and  capacity  of  the  circuit  and 
on  the  frequency  of  the  alternating  current  and  are  therefore  not  true  con- 
ductances. Both  of  these  terms  are  limited  chiefly  to  alternating  current 
circuits.  The  admittance  in  mhos  is  equal  to  the  effective  current  in 
amperes  divided  by  the  effective  electromotive  force  in  volts,  regardless 
of  what  the  phase  difference  may  be;  its  value  in  mhos  is  equal  to  the 
reciprocal  of  the  impedance  in  ohms.  It  is  also  equal  to  the  square  root 
of  the  sum  of  the  squares  of  the  conductance  and  the  susceptance,  all 
values  being  in  mhos.  The  susceptance  in  mhos  is  equal  to  the  wattless 
current  in  amperes  divided  by  the  electromotive  force  in  volts.  It  is 
also  equal  in  mhos  to  the  reactance  in  ohms  divided  by  the  sum  of  the 
squares  of  the  resistance  in  ohms  and  the  reactance  in  ohms.  For  further 
explanations  concerning  the  calculation  of  alternating  current  circuits 
reference  should  be  made  to  treatises  on  that  subject. 

The  only  unit  used  in  practice  is  the  mho,  which  is  the  reciprocal  of 
the  ohm ;  it  is  the  word  ohm  written  reversed  to  indicate  the  reciprocal. 
There  is  no  official  sanction  for  its  use,  but  as  there  is  no  other  practical 
unit,  it  has  come  into  use. 

The  electromagnetic  and  electrostatic  C.G.S.  units  are  the  re- 
ciprocals of  the  corresponding  units  of  resistance,  v  is  the  velocity  of  light. 

Logarithm 

1  CGS  unit  (elst)  =  109>2  mho.  About  %  X  10"11 12-045  7575 

=  1A-2CGS  unit  (elmg).  About  Vtt  X  lO"20.  21-045  7575 

1  mho=v2x  10~-'  CGS  units  (elst).  About  9X  1011 11-954  2125 

=  10~9  CGS  unit  (elmg) 9-000  0000 

1  megamho  =  1  000  000.  mhos 6-000  0000 

=  0.001  CGS  unit  (elmg) 3. 000  0000 

1  CGS  unit  (elmg)  =  v2  CGS  units  (elst).  About  9X1020..  20-954  2425 

=  109  mhos 9.000  0000 

=  1  000.  megamhos 3-000  0000 

The  relations  to  other  measures  are  as  follows,  f    (See  also  the  reciprocals 
of  those  under  resistance.) 
Mhos  =  1-^ohms. 

'     =   amperes-?- volts. 
'     =  watts  -r-  volts2. 
'*     =  amperes2-:- watts. 
Mega  mho  6  =  1  -^microhms. 
Mhos  conductance: 

=  1  -j-ohms  resistance,     For  direct  currents  only. 
=  amperes  energy  component  of  c ur re nt-J-  volts  total  e.m.f. 
=  watts  •*•  volts2. 

=  x/(nihos  admittance2  — mhos  susceptance2). 
=  VT(l-*-ohms  impedance2)  — mhos  susceptance2]. 
=  ohms  resistance-;- ohms  impedance2. 
=  ohms  resistance -v- (ohms  resistance2 -f- ohms  reactance2). 
=  ohms  resistance  X  mhos  admittance2. 
Mhos  susceptance: 

=  amperes  wattless  component  of  current -4- volts  total  e.m.f. 
=  X/(mhos  admittance2  — mhos  conductance2). 
=  ^[(1  -r-ohms  impedance2)  —  mhos  conductance2]. 
=  ohms  reactance-:- ohms  impedance2. 
=  ohms  reactance -r- (ohms  resistance2 -f  ohms  reactance2). 
=  ohms  reactance  X  mhos  admittance2. 

Mhos  admittance  =  total  effective  amperes -r- total  effective  volts. 
=  \/(mhos  conductance2-!- mhos  susceptance2). 
=  l-7-ohms  impedance. 

=  l-r-v/(°hms  resistance2  +  ohms  reactance2), 
^v^mhos  conductance -T- ohms  resistance). 
=  v/(mhos  susceptance -T- ohms  reactance). 

t  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  peri- 
odically varying  quantities. 


106     CONDUCTIVITY;  SPECIFIC  CONDUCTANCE. 


CONDUCTIVITY   [r] ;     SPECIFIC   CONDUCTANCE. 
(Conductance  X  length  -f-  cross-section  j    1  -v- resistivity .) 

These  units  are  used  to  measure  the  inherent  quality  of  a  material  to 
conduct  an  electric  current.  Conductivity,  which  is  the  reciprocal  of  resis- 
tivity, differs  from  conductance  in  that  the  latter  applies  to  a  given  circuit 
or  part  of  a  circuit,  and  its  amount  depends  on  the  length,  cross  section, 
and  quality  of  the  material;  while -conductivity  refers  only  to  the  nature 
of  the  material  and  is  always  the  same  for  the  same  material;  it  is  the 
conductance  of  a  unit  amount  of  the  material  like  a  cube  of  one  centimeter. 
It  bears  somewhat  the  same  relation  to  conductance  as  the  density  of  a 
material  does  to  the  weight  of  a  given  amount  of  it;  the  density  is  always 
the  same  for  that  material,  being  the  weight  of  a  unit  of  volume,  while  the 
total  weight  of  an  actual  piece  depends  on  the  size  of  that  piece.  As  the 
conductivity  is  a  quality  of  .a  material,  its  values  are  usually  given  in 
tables  of  physical  constants.  The  conductivity  may  also  be  calculated  from 
the  conductance  of  any  piece  or  part  of  a  circuit  by  multiplying  the  con- 
ductance in  mhos  (or  1  -f- resistance  in  ohms)  by  the  length  d,nd  dividing  by 
the  cross-section;  the  result  will  of  course  be  different  depending  upon  the 
units  used.  The  conductivity,  being  a  property  of  a  material,  is  the  same 
for  direct  and  for  alternating  current. 

The  values  of  the  conductivities  of  materials  are  not  in  general  use,  the 
reciprocal  quantity,  namely  resistivity,  being  generally  preferred,  as  it  is 
simpler  to  use  in  most  calculations.  The  use  Of  conductivities  in  practice 
is  generally  limited  to  electrolytes  and  for  making  comparisons  of  other 
conducting  materials  with  copper,  or  for  comparing  different  qualities  of 
copper  with  each  other. 

The  most  rational  unit  and  the  one  which  seems  to  be  coming  into  more 
general  use,  is  the  conductivity  of  a  material  of  which  a  cube  of  one  centi- ' 
meter  has  a  conductance  of  one  mho  between  two  parallel  sides.  It  will 
here  be  called  the  mho,  cubic  centimeter,  unit.  This  means  that  the 
resistance  of  a  column  of  that  material  one  centimeter  long  and  one  square 
centimeter  cross-section,  is  one  ohm,  that  is,  it  corresponds  to  the  ohm, 
cubic  centimeter,  unit  of  resistivity;  conductivities  stated  in  the  mho, 
cubic  centimeter,  unit  are  the  numerical  reciprocals  of  the  resistivities 
stated  in  the  ohm,  cubic  centimeter,  unit.  .This  unit  of  conductivity  is 
therefore  the  best  one  to  use  when  conductivities  are  to  be  converted  into 
resistivities  or  the  reverse.  This  unit  is  used  chiefly  for  electrolytes;  the 
best  conducting  aqueous  solutions  of  acids  at  40°  C.  have  a  conductivity  of 
about  one,  in  terms  of  this  unit. 

The  electromagnetic  C.G.S.  unit  is  the  same,  except  that  the  con- 
ductance is  stated  in  C.G.S.  units  instead  of  mhos.  Similarly  with  the 
electrostatic  C.G.S.  unit. 

The  conductances  in  mhos  of  a  wire  one  meter  long  and  one  square  milli- 
meter or  one  circular  millimeter  cross-section,  or  one  foot  long  and  one 
square  mil  or  one  circular  mil  cross-section,  may  also  be  used  as  units  of 
conductivity,  each  being  the  reciprocal  of  the  corresponding  unit  of  resis- 
tivity, which  see. 

The  most  usual  way  of  stating  the  conductivity  of  a  solid  material  is  to 
give  the  ratio  of  its  conductivity  to  that  of  pure  copper  as  a  standard;  this 
avoids  the  use  of  the  little-known  unit  of  conductance  (mho).  This  ratio 
is  usually  given  as  a  percent,  but  may  also  be  stated  as  so-and-so  many 
"copper  units"  of  conductivity.  The  Matthiessen  value  for  the  resistivity 
of  pure  copper  at  15°  C.  is,  according  to  Prof.  Lindeck  of  the  Reichsanstalt, 
1  687  in  microhm,  cubic  centimeter,  units.  The  conductivity  correspond- 


ties  greater  than  100%  when  based  on  the  Matthiessen  standard.  A  better 
value  for  pure  copper  is  the  one  used  as  standard  in  Germany,  presumably 
by  authority  of  the  Reichsanstalt,  which  according  to  Prof  Lindeck  is  a 
resistivity  of  1  667  in  microhm,  cubic  centimeter,  units,  at  15°  C.  The 


CONDUCTIVITY;  SPECIFIC  CONDUCTANCE.      107 

conductivity  corresponding  to  this  is  599  880.  in  mho,  cubic  centimeter, 
units.  In  the  following  table  this  is  called  the  copper  unit.  Mercury 
was  formerly  often  used  as  a  standard  of  comparison,  particularly  in  stat- 
ing the  conductivities  of  electrolytes.  The  resistivity  determined  from  the 
definition  of  the  international  ohm  is  94.073  4  in  microhm,  cubic  centimeter, 
units.  The  conductivity  corresponding  to  this  is  10  630.  in  mho,  cubic 
centimeter,  units.  In  the  following  table  this  is  called  the  mercury  unit. 

Logarithm 
1CGS  unit  (elst): 

10V*'2  mho,  cb.  cm,  unit.     About  %  X  10~n 15.045  7575 

=  94  073.4/7'2  mercury  unit.     About  1.045  26  X  10"10 16-019  2242 

l/7>2  COS  unit  (elmg).     About  Vo  X  lO"23 21-045  7575 

1  mho,  cb.  centimeter,  unit: 

v2X  10~9  CGS  units  (elst).    About  9X  1011 11-954  2425 

=  0.000  094  073  4  mercury  unit 5.973  4667 

=     0.000  001  687  copper  unit  (Matthiessen) 6-227  1151 

=     0.000  001  667  copper  unit 6-221  9356 

1C-9  CGS  unit  (elmg) 9. 000  0000 

1  mercury  unit: 

=  v2X  1.0630X10-5  CGS  units  (elst).    Ab.  9.567  000  X  1015  15-9807758 

10  630.  mho,  cb.  cm,  units 4-026  5333 

0.017  932  8  copper  unit  (Matthiessen) 2-253  6484 

0.017  720  2  copper  unit 2-248  4689 

=         0.000  010  630  CGS  unit  (elrng) 5-026  5333 

1  copper  unit  (Matthiessen): 

592  768.  mho,  cb.  cm,  units 5-772  8849 

=           55.763  7   mercury  units. 1-746  3519 

-0.000  592768   CGS  unit  (elmg) 4-772  8849 

1  copper  unit  =            599  880.  mho,  cb.  cm,  units 5-778  0644 

=           56.432  7   mercury  units 1-751  5311 

=  0.000  599  880  CGS  unit  (elmg) 4-778  0644 

1  CGS  unit  (elmg)  =             v2  CG,:*  units  (elst).    About  9X  102".  20-9542425 

109  mho,  cb.  cm,  units 9-000  0000 

=  94  073.4  mercury  units 4-973  4667 

=       1  687.  copper  units  (Matthiessen) 3-227  1151 

=       1  667.  copper  units 3-221  9356 

Conductivity  of  mercury  :  f 

=          10  630.  mho,  cb.  centimeter,  units 4-026  5333 

=  0.017  932  8  times  that  of  copper  (Matthiessen) 2-253  6484 

=  0.017  720  2  times  that  of  copper 2-248  4689 

Conductivity  of  copper  (Matthiessen):  t 

=  592  768.  mho,  cb.  centimeter,  units -  5-772  8849 

=  55.7637   times  that  of  mercury 1-746  3516 

Conductivity  of  copper:  § 

=  599  880.  mho,  cb.  centimeter,  units 5-778  0644 

=  56.432  7   times  that  of  mercury 1-751  5311 

The  relations  of  conductivity  to  other  measures  are  as  follows: 
Conductivity  (in  mho,  cb.  cm,  units). 

=  !•*•  resistivity  (in  ohm,  cb.  cm,  units). 
=  mhos  X  cm  length -r-sq.  cm  section. 
=  cm  length  -s-  (ohms  X  sq.  cm  section). 

t  Pure  mercury  at  0°  C. ;  based  on  the  definition  of  the  international  ohm. 
t  Matthiessen's  value  for  pure  copper  at  15°  C. ;   according  to  Prof.  Lin- 
deck. 

§  Pure  copper  at  15°  C. ;  according  to  Prof.  Lindeck. 


108  ELECTROMOTIVE    FORCE;   POTENTIAL. 


ELECTROMOTIVE  FORCE  [e.m.f.,  E,  e];  POTENTIAL} 
DIFFERENCE  OR  FALL  OF  POTENTIAL  [p.  d., 
U,  u];  STRESS;  ELECTRICAL  PRESSURE  ;  VOLT- 
AGE. (Magnetic  flux -f- time;  current  X  resistance.) 

These  units  are  used  to  measure  the  electrical  pressure,  stress,  or  motive 
force  which  produces  or  tends  to  produce  a  current,  just  as  pounds  measure 
the  pressure  of  air  either  in  the  form  of  compressed  air  or  wind,  or  as  the 
differences  of  level  of  water  measure  the  force  which  causes  the  water  to 
now  and  which  might  similarly  be  called  the  hydraulic  motive  force.  Ac- 
cording to  Ohm's  law  the  electromotive  force  in  volts  is  equal  to  the  current 
in  amperes  multiplied  by  the  resistance  in  ohms;  or  according  to  Joule's 
law  it  is  equal  to  the  power  in  watts  divided  by  the  amperes;  or  to  the 
square  root  of  ths  product  of  the  watts  and  tha  ohms;  these  apply  to  the 
electromotive  forces  or  differences  of  potential  of  direct  currents;  they 
apply  to  Alternating  current  electromotive  forces  also  when  there  is  no 
reactance  in  tho  circuit  and  therefore  no  phase  shifting  (caused  by  induct- 
ance or  capacity),  in  which  case  they  refer  to  the  effective  electromotive 
force;  when  th?re  is  re  >ctunce  in  such  circuits  the  electromotive  force  in 
volts  equals  the  current  in  amperes  multiplied  by  the  impedance  in  ohms. 

The  terms  electromotive  force  and  potential  are  used  synonymously 
in  practice,  although  the  use  of  the  latter  term  is  not  to  be  commended 
owing  to  its  more  general  meaning  in  physics.  Difference,  of  potential 
means  in  general  the  difference  betweon  two  absolute  potentials,  e.m.f.'s, 
or  potentials  in  general,  the  actual  values  of  which  need  not  be  known ; 
this  term  is  often  distinguished  from  electromotive  force,  in  that  the  latter 
applies  to  tho  total  whi  -h  is  generated  in  a  battery  or  dynamo  while  the 
difference  of  potential  applies  only  to  a  portion  of  it,  liko  that  available  at 
the  terminals,  or  that  between  any  two  pointe  on  a  circuit.  Voltage 
means  any  e.m.f.  or  difference  of  potential  when  expressed  in  volts.  Ab- 
solute potential  is  sometimes  used  to  denote  the  potential  above  or 
below  some  assumed  zero,  which  is  usually  taken  as  that  of  the  earth. 

The  unit  universally  used  is  the  volt,  by  which  is  here  meant  the  inter- 
national voit  of  the  International  Congress  of  1893  in  Chicago,  denned  as 
that  electromotive  force  which  will  maintain  one  international  ampere 
through  one  international  ohm,  represented  for  practical  purposes  by 
1-^-1 .434  of  that  of  a  Clark  cell  at  15°  C.f  It  was  made  legal  in  this  country 
by  Congress  in  1894,  and  is  adopted  by  the  National  Bureau  of  Standards. 
The  "saturated"  Weston  or  cadmium  standard  cell  (with  excess  of 
crystals)  may  eventually  be  substituted  for  the  Clark  cell  as  the  official 
standard  because  it  has  a  much  smaller  temperature  coefficient  (see  table 
below);  the  relation  between  the  Clark  and  this  Weston  cell  being  known 
quite  definitely  (see  table  below),  either  may  be  used  as  the  standard;  in 
this  table  this  ratio  and  the  value  of  the  Clark  cell  are  used  as  the  funda- 
mental values.  There  is  another  type  of  cadmium  cell  called  the  '•  unsatu- 
rated  "  Weston  cell,  in  which  there  is  no  excess  of  crystals  at  ordinary 
temperatures,  as  the  solution  is  saturated  at  4°  C. ;  this  has  the  advantage 
of  having  a  still  lower  temperature  coefficient,  which  can  be  neglected  en- 
tirely at  ordinary  temperatures;  the  National  Bureau  of  Standards  does 
not  regard  it  safe  to  assign  a  definite  value  to  this  unsaturated  Weston  cell 
owing  to  ths  possibility  of  the  seal  being  imperfect  and  the  consequent 
change  in  the  concentration  of  the  solution,  and  also  the  impossibility 
of  ascertaining  the  exact  temperature  at  which  the  solution  was  saturated. 
The  e.m.f.  of  this  cell,  with  a  solution  saturated  at  4°  C.,  is,  however,  1.019  8 
international  volts,  this  value  being  the  same  as  that  of  the  saturated  cell 
at  the 'same  temperature.  A  number  of  such  cells  belonging  to  the  Bureau 
have  been  intercompared;  and  were  found  to  differ  by  a  number  of  units 

t  This  is  the  value  of  the  volt  used  in  calibrating  the  Weston  voltmeters. 


ELECTROMOTIVE    FORCE;    POTENTIAL.  109 

in  the  last  decimal  place;  hence  the  cell  should  not  be  employed  as  a  stand- 
ard of  reference,  although  as  a  working  standard  it  can  hardly  be  improved 
upon. 

The  value  adopted  at  the  Reichsanstalt  for  the  electromotive  force  of  thb 
Clark  cell  is  based  upon  a  determination  of  its  e.m.f.  in  terms  of  the  elec- 
trochemical equivalent  of  silver  and  the  unit  of  resistance,  and  also  upon  i* 
similar  determination  of  the  e.m.f.  of  the  Weston  or  cadmium  cell,  togethev 
with  a  determination  of  the  ratio  of  the  values  of  these  two  cells.  As  the 
values  thus  obtained  for  the  Clark  and  Weston  or  cadmium  cells  by  the 
silver  voltameter  did  not  agree  with  the  directly  determined  ratio,  each  of 
the  silver  voltameter  determinations  was  given  equal  weight  and  the  two 
separate  values  adjusted  so  as  to  give  the  ratio  directly  determined.  The 
Reichsanstalt's  value  thus  obtained  for  the  Clark  cell  is  1.432  85  instead  of 
1.434  as  defined  by  the  International  Congress  and  legal  in  this  country,  f 
This  Reichsanstalt  value  may  be  more  accurate,  but  is  not  legalized  here. 
As  the  cells  are  the  same,  this  makes  a  very  slight  difference  between  the 
volt  used  by  the  Reichsanstalt  and  that  legal  and  used  in  this  country  (the 
international  volt).  The  National  Bureau  of  Standards  uses  as  the  funda- 
mental units  those  of  resistance  and  electromotive  force,  obtaining  the 
ampere  from  them,  thus  bringing  all  three  into  agreemen'  with  each  other. 
According  to  Weston  the  international  concrete  volt,  an^ore,  and  ohm, 
as  defined  by  the  Chicago  Congress,  agree  with  each  other. 

In  some  of  the  relations  with  other  units,  such  as  the  absolute,  the  mag- 
netic, the  energy  units,  etc.,  the  value  of  the  international  volt  as  above 
defined  in  terms  of  the  Clark  cell  is  in  the  following  tables  assumed  to  be 
equal  to  the  theoretical  value,  namely  108  electromagnetic  C.G.S.  units, 
which  value  is  sometimes  called  the  true  volt.  According  to  the  theoret- 
ical definition  of  a  volt,  it  is  the  difference  of  potential  generated  in  a  con- 
ductor which  cuts  10s  C.G.S.  units  of  magnetic  flux  (or  10s  maxwells  or 
lines  of  force)  per  second;  or  it  is  the  difference  of  potential  generated  per 
centimeter  length  of  a  conductor  moving  transversely  through  a  magnetic 
field  of  a  density  of  1  C.G.S.  unit  of  density  (or  1  gaiiss)  at  a  velocity  of 
10s  centimeters  (or  1  000  kilometers)  per  second.  A  difference  of  potential 
thus  generated  is  moreover  directly  proportional  to  the  amount  of  magnetic 
flux  traversed  per  second.  In  the  older  literature  the  e.m.f.  of  a  Daniell 
cell  (about  1.1  volt)  was  often  used  as  a  unit. 

The  electromagnetic  C.G.S.  unit  (or  absolute  unit)  is  the  difference 
of  potential  generated  at  the  ends  of  a  conductor  1  centimeter  long  moving 
through  a  magnetic  field  of  unit  density  (one  gauss)  at  a  speed  of  1  centi- 
meter per  second  perpendicularly  to  the  direction  of  the  field;  or  more 
briefly,  it  is  the  difference  of  potential  induced  in  a  conductor  which  cuts 
one  unit  of  magnetic  flux  (one  maxwell  or  one  line  of  force)  per  second. 

The  electrostatic  C.G.S.  unit  (or  absolute  unit)  is  that  difference  of 
potential  through  which  one  electrostatic  unit  of  quantity  falls  when  the 
work  done  by  it  is  one  erg 

ELECTROMOTIVE   FORCE. 

**  Accepted  by  the  National  Bureau  of  Standards. 

*  Checked  by  Dr.  Frank  A.  Wolff,  Jr.,  Asst.  Phys.  National  Bureau  of 
Standards. 

Aprx.  means  within  2%.  By  "volt"  is  meant  international  volt  unless 
otherwise  stated,  v  is  the  velocity  of  light. 

Logarithm 

1  t  GS  unit  (elmg)  =       1  abvolt 0-000  0000 

=  0.01  microvolt 2-000  0000 

=  10-«  voit g-000  0000 

=  1/v  CGSunit(elst).    About ^ XIO"10.  ..    II. 522  8787 

1  abvolt  =  1  CGS  unit  (elmg) 0-000  0000 

1  microvolt  =  100.  CGS  units  (elmg) 2-000  0000 

=  0.000  001   volt 6-000  0000 

1  millivolt  =  0.001  volt 3-000  0000 

1  legal  volt  =0.997  178*  volt.    Aprx.  1 1-998  7726 

1    ampere  X  1  legal  ohm. 

t  The  difference  corresponds  almost  exactly  to  that  due  to  one  Centigrade 
degree  difference  of  temperature;  it  is  about  8  hundredths  of  one  percent 


110     ELECTROMOTIVE  FORCE;  POTENTIAL. 

1  volt  [V,  v] :  Logarithm 

10s     CGS  units  (elmg) 8-0000000 

=  1  000  000.     microvolts .  .  .      6-000  0000 

=         1  000.     millivolts 3-000  0000 

=    1.002  83*    legal  volts.     Aprx.'l 0  001  2274 

=  0.999  198**  volt  of  Reichsanstalt. t  Aprx.  subtr.  8/ioo%..  1-999  6518 
=  0.980962**  Weston  (satur.)  cell  at  20°  C.  Ap.sub.2%...  1-9916523 
=  0.980567  Weston  (unsat.)  cell  at  any  temp.t  Aprx. 

subtr.  2% 1.991  4774 

=  0.697  350*    Clark  cell  at  15°  C.    Aprx.  i/w 1.843  4508 

108 /v    CGS  unit  (elst).    About  Men 3-522  8787 

0.001    kilovolt f-000  0000 

1  international  volt  =  1  volt,  which  see  above ,      0  000  0000 

1  true  volt  =  10s  CGS  units  (elmg) 8-000  0000 

1  volt  of  Reich8aiistalt  =  1.000  803**  volts.f  Ap.  add 8/loo%.      0-000  3484 
1  Weston  (cadmium;  saturated)  cell  at  2O°  C.  with  exce.ss  of  crystals: 

=    1.019  4**  volts.    Aprx.  add  2% 0  008  3477 

=    1.018  6**  volts  of  Reichsanstalt.    Aprx.  add  2% 0  007  9993 

=  0.71088*    Clark  cell  at  15°  C.    Aprx.  % 1-8517985 

1  Weston  (cadmium;  unsaturatedf)cell  at  any  ordinary  temperature: 

=  1 .019  8*  volts.     Aprx.  add  2% 0  008  5226 

=  1.019  0*  volts  of  Reichsanstalt.    Aprx.  add  2% 0  008  1742 

1  Daiiiell  cell  =  1.1  volts  approximately  (unreliable) 0-041  3927 

1  Clark  cell  at  15°  C.: 

=         1.434**  volts.     Aprx.  1% 0-1565492 

=  1.432*85**  volts  of  Reichsanstalt.    Aprx.  *% 0-156  2008 

=      1.4O6  T**  Weston  cells  at  20°  C     Aprx.  % 0-1482015 

1  CGS  unit  (elst)=  v  CGS  units  (elmg).     Ab.  3X1010.    10-477  1213 

=  v  X  10-<s  volts.     About  300 2-477  1213 

1  abstavolt 0  000  0000 

=  vX  10~n  kilovolt.     About  s/lo I  477  1213 

1  abstavolt  =  1  CGS  unit  (elst) 0  000  0000 

1  kilovolt  =  1000.      volts 3-000  0000 

=   Wn/v  CGS  units,  (elst).    About  io/3 0-522  8787 

1  meg-avolt  =  1  000  000.  volts 6  000  0000 

1  absolute  unit  -  1  CGS  unit,  either  elmg  or  elst 0-000  0000 

The  relations  of  volts  to  other  measures  are  as  follows:  J 
Volts  —  amperes  X  ohms. 

.  =  ohms  X  coulombs  -=-  seconds. 
=  watts  -r-  amperes. 
=  kilowatts  X  1  000.  -T-  amperes. 
=-  >/(  watts  X  ohms). 
=  watts  X  seconds  -5-  coulombs. 
=  joules  -r-  coulombs. 
=  joules  -r-  (amperes  X  seconds). 
=  \/(  joules  X  ohms -r- seconds). 
=  coulombs  -j-  farads. 
=  coulombs  X  1  000  000.  ^-microfarads. 
=  \/( joules  of  stored  energy  X  2  -4- farads). 
=  1  000.  X  \/( joules  of  stored  energy  X  2  -5- microfarads). 
=  maxwells  X  number  of  turns  -*-  (seconds  X  1 08). 
=  gausses  X  sq.  centimeters  -J-  (seconds  X  108). 
Induced  volts: 

=  henrys  X  rate  of  change  of  amperes  per  second. 

=  time  constant  in  seconds  X  ohms  X  rate  of  change  of  amperes  per  sec. 
Applied  volts: 

=  henrys  X  final  amperes -Mime  constant  in  seconds. 

=  joules  of  kinetic  energy  of  the  current  X  ohms  X  2  -T- (henrys  X  final 

amperes). 

=  ohm sX\/[ joules  of  kinetic  energy  of  the  current  X  2 -r-henrys  J. 
=  joules  of  kinetic  energy  of  the  current  X  2  -*-( time  constant  in  sec- 
onds X  final  amperes). 

=  >/[( joules  of  kinetic  energy  of  the  current  X  ohms  X  2)  •*•  time  con- 
stant in  seconds  . 

1  See  explanatory  note  above. 

}  Consult  treatises  on  alternating  currents  for  the  limiting  conditions. 


ELECTROMOTIVE  FORCE;  POTENTIAL. 


Ill 


For  alternating  current  circuits:  f 

Volts  =  watts -r- (am  peresX  cos  ^J) 
=  \/[( watts  Xohrns)-r- cos  <£tl 
=  amperes-f-(faradsXfrequencyX6.283  19§). 
=  amperes  X  1  000  000.  -*-  (microfarads  X  frequency  X  6.283  19§  ). 

Induced  volts: 

=  \/[(apph'ed  volts)2— (amperes X ohms  resistance)2]. 
=  henrys  X  amperes  X  frequency  X  6.283  19§ . 

=  time  constant  in  seconds  X  ohms  resistance  X  amperes  X  frequency  X 
6.283  19§. 

Applied  volts: 

=  VT(amperesXohms  resistance)2 -(-(induced  volts)2], 
amperes  X  x/[(henrys  X  frequency  X  6.283  19§  )2  +  (ohms  resistance)2] . 


=  amperes  X  ohms     resistance  X  \/[(time 
quencyX  6.283  19§)2  +  1J. 


constant     in     seconds  X  f  re- 


Form  ul  as  for  the  temperature  corrections  in  Centigrade  de- 
grees, between  0°  and  30°,  determined  by  the  Reichsanstalt  and  accepted 
by  the  National  Bureau  of  Standards.  They  apply  equally  well  to  the 
international  values  and  to  the  Reichsanstalt  values,  t  is  the  temperature 
in  Centigrade  degrees;  Et  is  the  electromotive  force  at  that  temperature; 
while  Ei5  and  E-M  are  the  standard  values  at  1,5°  and  20°  respectively 
as  given  in  the  above  table. 

For  the  Clark  cell: 

^  =  #15-0.001  19  (t- 15) -0.000  007  (<-15)2.  ** 

For  the  Weston  (saturated)  cell: 

Et  =  E20  -  0.000  038  (t  -  20)  -  0.000  000  65  (t  -  20)2.  ** 


K.M.F.   of  Clark    and   Weston    Cells  at  Different   Temperatures 

Calculated  from  these  Formulas. 


Clark. 

Weston  (saturated). 

c° 

F°. 

International 

Reichsanstalt 

International 

Reichsanstalt 

Volts. 

Volts. 

Volts. 

Volts. 

0 

32.0 

1.4503 

1.4491  2 

.0199 

.0191 

10 

50.0 

1.4398 

1.43862 

.0197 

.0189 

13 

55.4 

1.4364 

.  435?  0 

.0196 

.0188 

14 

57.2 

1.4352 

.43403 

.0196 

.0188 

15 

59.0 

1.4340 

.4338  5 

.0196 

.0188 

16 

60.8 

1.4328 

.43165 

.0195 

.0187 

17 

62.6 

1.4316 

.43044 

.0195 

.0187 

18 

64.4 

1.4304 

.42922 

.0195 

.0187 

19 

66.2 

1.4291 

.42798 

.0194 

.0186 

20 

68.0 

1.4279 

.4267  3 

.O194 

.0180 

21 

69.8 

1  .4266 

.42546 

1.0194 

.0186 

22 

71.6 

1.4253 

.4241  8 

1.0193 

.0185 

23 

73.4 

1.4240 

.42288 

1.0193 

.0185 

24 

75.2 

1  .  4227 

1.42157 

1.0192 

.0184 

30 

86.0 

1.4146 

1.41343 

1.0190 

1.0182 

t  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  peri- 
odically varying  quantities. 

J  </>  is  the  phase  difference  in  degrees. 

§  Or  2*.     Aprx.  ^XIO.     Log.  0.798  1799. 


112 


ELECTRICAL   CURRENT. 


ELECTRICAL  CURRENT  [I,  i];  CURRENT  STRENGTH 
or  INTENSITY.  (Electromotive  force  ~  resistance; 
quantity  -v-  time. ) 

These  units  are  used  to  measure  the  rate  of  flow  or  passage  of  units  of 
electricity  per  second,  just  as  the  rate  of  flow  of  water  or  air  is  measured  in 


„  --  , !  power  in  watts  divided  by  t 

volts;  or  to  the  square  root  of  the  quotient  of  the  watts  divided  by  the 
ohms.  These  anply  to  direct  currents  without  counter-electromotive  forces ; 
they  apply  to  alternating  currents  also  when  there  is  no  reactance  in  the 
circuit  (caused  by  inductance  or  capacity),  in  which  case  they  refer  to  the 
effective  current.  In  any  alternating  current  circuit  with  reactance  the 
current  in  amperes  is  equal  to  the  electromotive  force  in  volts  divided  by 
the  impedance  in  ohms.  An  ampere  is  also  equal  to  a  passage  of  one 
coulomb  per  second. 

The  unit  universally  used  is  the  ampere,  by  which  is  here  meant  the 
international  ampere  of  the  International  Congress  of  1893  at  Chicago, 
defined  as  equal  to  Vio  of  the  C.G.S.  electromagnetic  unit  of  current;  it  was 
made  legal  in  this  country  by  Congress  in  1894.  For  practical  purposes  it 
is  defined  by  that  congress  as  the  current  which  (under  specified  conditions) 
deposits  0.00 1  118  gram  of  silver  per  second. f  The  ampere  is,  however, 
preferably  determined  from  the  ohm  and  the  voltage  of  a  standard  cell. 
Owing  to  the  slight  discrepancy  in  the  units  of  current,  electromotive  force, 
resistance,  and  Ohm's  law,  the  National  Bureau  of  Standards  has  selected 
two  of  these  as  the  fundamental  units,  namely  those  of  resistance  and  elec- 
tromotive force  established  by  the  International  Congress  and  legalized  in 
this  country,  and  from  these  the  ampere  is  derived,  thus  bringing  all  three 
into  agreement  with  Ohm's  law.  Individual  Clark  standard  cells  agree 
with  each  other  to  within  at  least  2  parts  in  10  000.,  and  by  the  use  of  care- 
fully purified  materials  this  agreement  can  be  still  closer,  while  different 
determinations  of  the  electro-chemical  equivalent  of  silver  differ  by  con- 
siderably larger  amounts,  unless  repeated  under  perfectly  definite  condi- 
tions and  made  with  great  care.  The  Reichsaristalt  measured  the  electro- 
motive force  of  the  Clark  cell  in  terms  of  the  ohm  and  the  ampere  based 
on  the  silver  voltameter,  but  obtained  a  slightly  different  value  for  this  cell 
from  that  defined  by  the  International  Congress;  hence  the  ampere  of  the 
Reichsanstalt,  which  is  in  agreement  with  the  volt  and  ohm  there  used,  is 
slightly  different  from  the  ampere  used  in  this  country,  which  is  based  on 
the  international  volt  and  ohm,  although  both  these  amperes  were  originally 
intended  to  be  the  same.  But  this  discrepancy,  which  is  only  about  8  him- 
dredths  of  one  percent,  is  quite  negligible  in  ordinary  practice.  According 
to  Weston  the  concrete  volt,  ampere,  and  ohm  as  defined  by  the  Chicago 
Congress  are  in  agreement  with  each  other  to  within  one  pan  in  1 000. 

In  some  of  the  relations  with  other  units,  such  as  the  absolute,  the  mag- 
netic, the  energy  units,  etc.,  the  value  of  the  international  ampere  as  above 
defined  is  in  the  following  tables  assumed  to  be  equal  to  the  theoretical 
value,  namely  Vio  of  the  electromagnetic  C.G.S.  unit,  which  value  is  some- 
times called  the  true  ampere. 

The  electromagnetic  C.G.S.  unit  (or  absolute  unit)  is  that  current 
which,  flowing  in  the  circumference  of  a  circle  of  one  centimeter  radius, 
will,  for  every  centimeter  length  of  circumference,  exert  in  air  a  force  of  one 
dyne  on  a  unit  magnetic  pole  placed  at  the  center;  one  whole  circumfer- 
ence therefore  exerts  a  force  of  2n  dynes  on  that  pole.  Or  under  the  same 
conditions,  every  centimeter  of  the  circumference  will  in  air  produce  at  the 
center  a  magnetic  field  of  one  unit  density  (one  gauss),  that  is,  one  unit  of 
magnetic  flux  (one  maxwell)  per  square  centimeter. 

The  electrostatic  C.G.S.  unit  (or  absolute  unit)  is  the  current  which 
flows  when  one  electrostatic  C.G.S.  unit  of  quantity  passes  per  second. 

f  This  is  the  value  used  in  calibrating  the  Weston  amperemeters. 


ELECTRICAL  CURRENT.  113 

ELECTRICAL  CURRENT. 

**  Accepted  by  the  National  Bureau  of  Standards. 

*  Checked  by  Dr.  Frank  A.  Wolff,  Jr.,  Asst.  Phys.  National  Bureau  of 
Standards. 

Aprx.  means  within  2%.  By  "ampere"  is  meant  the  international  am- 
pere, assumed  to  be  equal  to  the  international  volt  divided  by  the  interna- 
tional ohm,  unless  otherwise  stated,  v  is  the  velocity  of  light. 

Logarithm 

1  CGS  unit  (elst)  =        1    abstatampere 

=  107/v  microampere.    About  %  -r- 1  000 

=  10/v  ampere.    About  \i  X  10~9 

=     l/v  CGS  unit  (elmg).    About  Yz  X  lO"10.. . 

1  abstatampere  =  1  CGS  unit  (elst) 

1  microampere  =          v/107  CGS  units  (elst).    About  3  000.  . 

=  0.000  001   ampere 

1  milliampere  =v/10  000  CGS  units  (elst).    About  3  000  000. 

=        0.001  ampere 

=     0.0001  CGS  unit  (elmg) '. 

1  ampere  [A,  a]=          v/10      CGS  units  (elst).    About  3  X  10a . 

=  1000.     milliamperes 

=  0.999  198**  ampere  of  Reichsanstaltf 

0.1       CGS  unit  (elmg) 

1  international  ampere  =  1  ampere,  which  see  above 

1  true  ampere  =0.1  CGS  unit  (elmg) 

1  ampere  of  Reichsai^talt  -  1.000  803**  amperesf 

1  ampere  of  Nat.  1$ urea. a    of  Standards  =  1  volt-r- 1  ohm. 

1  CGS  unit  (elmg)  =    v  CGS  units  (elst).    About  3 X  10  ° 10-477  1213 

=  10  amperes 1 .000  0000 

=    1   absampere 0  000  0000 

1  absampere  =  1  CGS  unit  (el Tig) 0-000  0000 

1  kiloampere  =  1  000.  amperes 3  000  0000 

=     100.  CGS  units  (elmg) 2-000  0000 

1  absolute  unit  =  l  CGS  unit,  either  elst  or  elmg 0-000  0000 

The  relations  to  other  measures  are  as  follows:  J 
Amperes  =  volts  -f-  ohms 

=  coulombs  -f-  seconds. 
=  watts  -r-  volts. 
=  1  000  X  kilowatts  -4-  volts. 
=  \/(  watts  -r-  ohms). 
=  joules -H(  volts  X  second  ). 
"          =\/[joules-:- (ohms  X  seconds)]. 

'flute  of  change  of  amperes  per  second: 

=  induced  volts  -r-  henrys. 

=  induced  volts  -5-  (ohms  X  time  constant  in  seconds). 

l'i;il  amperes: 

=  applied  volts X  time  constant  in  seconds -4- henrys. 

--=  joules  of  kinetic  energy  of  the  current X ohms X 2 ^-(henrys X applied 
volts). 

=  \/(  joules  of  kinetic  energy  of  the  current  X  2  -s-  henrys). 

=  joules  of  kinetic  energy  of  the  current  X  2  -s-  (applied  volts  X  time  con- 
stant in  seconds). 

=  vT Joules  of  kinetic  energy  of  the  current  X  2  -5-  (ohms  X  time  constant 
in  seconds)  . 


f  See  explanatory  notes  above. 

j  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  period- 
ically varying  quantities 


114   ELECTRICAL  CURRENT;  CURRENT  DENSITY. 

When  the  flux  is  due  only  to  the  current,  as  in  self-induction,  and  when 
there  is  no  magnetic  leakage: 

Final  amperes: 

==  (max well sX number  of  turns) -:- (henrys  X  10s). 

=  (maxwells  X  number  of  turns)  •*-  (time  constant  in  seconds  X  ohms  X 

10s). 

=  ergs  of  kinetic  energy  6f  the  current  X  20  -«-  (maxwells  X  no.  of  turns). 
=  joules  of  kinetic  energy  of  the  current  X  2  X  10s  -?-  (maxwells  X  number 

of  turns). 

When  there  is  magnetic  leakage,  substitute  in  the  above  for  the  quantity 
"max wells X  number  of  turns,"  the  mean  flux  turns,  that  is  the  "mean 
max  wells  X  number  of  turns."     Thus: 
Final  amperes: 

=  (mean  maxwells  X  number  of  turns)  -r-  (henrys  of  self-induction  X  10s). 

When  the  flux  is  from  an  external  source,  and  independent  of  the  current 
as  in  mutual  induction,  and  when  there  is  no  magnetic  leakage: 
Final  amperes: 

=  ergs  of  kinetic   energy  of  the  current  X  10  -5-  (maxwells  X  no.  of  turns). 

=  joules  of  kinetic  energy  of  the  current  X  10s  -s-  (maxwells  X  no.  of  turns) 

When  there  is  magnetic  leakage,  make  the  same  substitution  as  described 
above. 
Final  amperes  in  primary: 

=  mean   maxwells   through   secondary X secondary   turns-:- (henrys   of 
mutual  induction  X  10s). 

For  alternating-current  circuits  f: 
Amperes: 

=  watts  -4-  (volts  X  cos  $  J  ). 
=  >/[ watts  -s-  (ohms  X  cos  0J)]. 

=  \/[applied  volts2  — induced  volts2] -r-ohms  resistance. 
— farads X volts X frequency  X 6. 283  19  §. 
=  microfarads  X  volts  X  frequency  X  6.283  19  §  •*•  1  000  000. 
=  induced  volts -s- (henrys  X  frequency X6.283  19  §). 

=•  induced    volts •+•  (time    constant   in    seconds  X  ohms    resist anceX  fre- 
quency X  6.283  19§). 
^applied  volts -i-VIX  henrys  X frequency X 6. 283  19  §)2+(ohms  res.)2J. 

CURRENT    DENSITY.     (Current -surface.) 

These  units  are  used  to  measure  the  current  flowing  through  a  unit  cross- 
section  of  a  wire  or  other  conductor;  or  the  amount  of  current  flowing  into 
or  out  of  a  unit  surface  of  an  electrode  in  an  electrolyte. 

Aprx.  means  within  2%. 

1  ampere  per  sq.  meter:  Logarithm 

=      0.092  903  4  ampere  per  sq.  foot.      Aprx.  H12  -*-10 29680317 

=•=                   0.01  ampere  per  sq.  decimeter 2-000  0000 

=  0.000  645  163  ampere  per  sq.  inch.     Aprx.  Vn  -*- 1  000 4-809  6692 

1  ampere  per  sq.  foot: 

=       10.763  87  amperes  per  sq.  meter.     Aprx  12/n  X  10 1  031  9683 

=   0.107  6387  ampere  per  sq.  decimeter.    Aprx.i%i  -5-100...  1-0319683 

=  0.006  944  44  ampere  per  sq.  inch.     Aprx.  7/iooo 3-841  6375 

1  ampere  per  sq.  decimeter: 

100  amperes  per  sq.  meter 2-000  0000 

=     9.290  34  amperes  per  sq.  foot.    Aprx.  11/12  X  10 0  968  0317 

=  0.064  5163  ampere  per  sq.  inch.     Aprx.Vii-^10 2-8096692 

t  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  peri- 
odically varying  quantities. 

|  0  is  the  phase  difference  in  degrees. 

§  Or  2*.     Aprx.  ^iX  10.    Log  0-798  1799- 


CURRENT  DENSITY;    ELECTRICAL  QUANTITY.  115 

Logarithm 
1  ampere  per  sq.  inch: 

=    1  550.00  amperes  per  sq.  meter.    Aprx.  ity  X  1  000 3. 190  3308 

=  144.  amperes  per  sq.  foot.    Aprx.  Vr  X  1  000 2-158  3625 

=    15.500  0  amperes  per  sq.  decimeter.     Aprx.  ity  X  10.  .  .    1-1903308 

=  0.785  398  ampere  per  circular  inch.    Aprx.  s/io 1-895  0899 

=  0.155  000  ampere  per  sq.  centimeter.    Aprx.  2/ia 1-190  3308 

=  0.121  736  ampere  per  circular  cm.    Aprx.  12-5-100 1-085  4207 

1  ampere  per  circular  inch: 

=    1.273  24  amperes  per  sq.  inch.    Aprx.  *% 0-104  9101 

=  0.197  352  ampere  per  sq.  centimeter.    Aprx.  2Ao 1-295  2409 

=  0.155  000  ampere  per  circular  cm.    Aprx.  %s 1-190  3308 

1  ampere  per  sq.  centimeter: 

=    929.034  amperes  per  sq.  foot     Aprx.  HiaXl  000 2-968  0317 

=  100.  amperes  per  sq.  decimeter 2  000  0000 

=   6.451  63  amperes  per  sq.  inch.    Aprx.  6^2 0  809  6692 

=   5.067  09  amperes  per  circular  inch.    Aprx.  5 0-704  7591 

=  0.785398  ampere  per  circular  centimeter.    Aprx.%o.  ...    1-8950899 

1  ampere  per  circular  centimeter: 

=  8.214  47  amperes  per  sq  inch.    Aprx.  %  X  10 0-914  5793 

=  6.451  63  amperes  per  circular  inch.    Aprx.  *% 

=  1.273  24  amperes  per  sq.  centimeter.    Aprx.  *% 

1  ampere  per  sq.  millimeter: 

=        0.785  398  ampere  per  circular  mm.    Aprx.  8/io 

=  0.000  645  163  ampere  per  sq.  mil.      Aprx.  V\i  -*- 1  000.  .  . 
=  0.000  506  709  ampere  per  circular  mil.    Aprx.  >£-*•  1  000  . 

1  ampere  per  circular  millimeter: 

1.273  24  amperes  per  sq.  millimeter.    Aprx.  '%.... 

=  0.000  821  447  ampere  per  sq.  mil.    Aprx.  M3  -5- 100 

=  0.000  645  163  ampere  per  circular  mil.     Aprx.  7/n  •*•  1  000 

1  ampere  per  sq.  mil: 

=  1  550.00  amperes  per  sq.  millimeter.  Aprx.  *#  X  1  000.. 
=  1  217.36  amperes  per  circular  mm.  Aprx.  %  X  1  000.  .  . 
=  0.785  398  ampere  per  circular  mil.  Aprx.  9io 

1  ampere  per  circular  mil: 

=  1  973.52  amperes  per  sq.  millimeter.    Aprx.  2  000 3-295  2409 

=  1  550.00  amperes  per  circular  mm.    Aprx.  Hfr  X  1  000  .  .  .    3-190  3308 
=  1.273  24  amperes  per  sq.  mil.    Aprx.  1% 0-104  9101 


ELECTRICAL   QUANTITY  [Q,  q]j    CHARGE.       (Current 
X  time ;    capacity  X  electromotive  force. ) 

These  units  are  used  to  measure  the  amount  of  electricity  as  such,  just 
as  a  quantity  of  matter  might  be  measured  in  the  number  of  units  called 
molecules,  which  it  contains.  The  number  of  units  of  electricity  in  a 
given  quantity  of  electricity  is  the  same  Avhether  or  not  it  is  flowing  in 
the  form  of  a  current,  or  whether  or  not  it  is  subjected  to  an  electrical 
pressure,  just  as  the  number  of  molecules  in  a  given  weight  of  air  is  the 
same  whether  or  not  it  is  in  motion,  as  in  a  wind,  or  whether  or  not  it  is 
under  pressure,  as  in  compressed  air.  The  quantity  of  electricity  in 
coulombs  is  equal  to  the  current  in  amperes  multiplied  by  the  time  in 
seconds;  or  to  the  energy  in  joules  divided  by  the  voltage;  or  to  voltage 
multiplied  by  the  capacity  of  a  condenser  in  farads. 

The  unit  generally  used  is  the  coulomb,  by  which  is  here  meant  the 
international  coulomb  of  the  International  Congress  of  1893  at  Chicago, 
denned  as  "the  quantity  of  electricity  transferred  by  a  current  of  one 
international  ampere  in  one  second."  It  was  made  legal  in  this  country  by 
Congress  in  1894,  and  is  accepted  by  the  National  Bureau  of  Standards. 
For  practical  purposes  it  is  that  quantity  which  will  deposit  0.001  118 
gram  of  silver  in  a  silver  voltameter.  In  some  of  the  relations  with 
other  units,  such  as  the  absolute,  the  energy  units,  etc.,  the  above  value 
is  in  the  following  tables  assumed  to  be  equal  to  the  theoretical  value, 
namely  ^4o  the  electromagnetic  C.G.S.  unit.  Another  unit  frequently 
used,  especially  with  batteries  and  in  electrochemistry,  is  the  ampere- 


116  ELECTRICAL    QUANTITY;    CHARGE. 

hour;    it  is  the  quantity  of  electricity  transferred  by  a  current  of  one 
ampere  in  one  hour. 


in  one  second.  ine  B*B«j»ru»w»«w  \;.VJT. .-».  unii<  ^ur  ausomie  uiui;  is 
that  quantity  of  electricity  which  in  air  exerts  a  force  of  one  dyne  on  an 
equal  quantity  one  centimeter  distant. 

ELECTKICAL  QUANTITY;    CHARGE. 

Aprx.  means  within    2%.     By  "coulomb"  is  meant  the   international 

coulomb,     v  is  the  .velocity  of  light. 

Logarithm 

1  CGS  unit  [elst]: 

1  abstatcoulomb 0  000  0000 

=           107/y  microcoulomb.     About  J^-s-  1  000 4  522  8787 

=             10/v  coulomb.     About  MX  10~» 15-522  8787 

=               l/v  CGS  unit  (elmg).     About  ^X  10~10 11-5228787 

=  1-4- (360  v)  ampere-hour.     About  9.259 X  10"14 14-9665762 

1  abstatcoulomb  =  1  CGS  unit  (elst) 0-000  0000 

1  microcoulomb: 

=    vX  0.000  000  1  CGS  units  (elst).     About  3  000.. 3-4771213 

=              0.000  001  coulomb 6-000  0000 

=           0.000  000  1  CGS  unit  (elmg) 7-000  0000 

=  2.777  78  X  10-10  ampere-hour.     Aprx.  !MX  10"10 10-443  6975 

1  coulomb  [C,  c] : 

v/10  CGS  units  (elst).     About  3  000  000  000...  .  Q.477  1213 

=         1  000  000.  microcoulombs 6  000  0000 

=                     0.1  CGS  unit  (elmg) 1-000  0000 

=  0.000  277  778  ampere-hour.     Aprx.  "4+  10  000 4  443  6975 

1  international  coulomb  =  1  coulomb,  which  see  above. 

1  true  coulomb -0.1  CGS  unit  (elmg) 1-000  000^ 

1  ampere-second  ==  1  coulomb,  which  see  above. 

1  CGS  unit  (elmg): 

=                      1  abscoulomb 0  000  0000 

=                      v  CGS  units  (elst).     About  3  X  1010 10-447  1213 

=                    10  coulombs 1-000  0000 

=  0.002777  78  ampere-hour.     Aprx   ^-s-1  000 3-443  6975 

1  abscoulomb  =  1  CGS  unit  (elmg) 0  000  0000 

1  ampere-hour  [ah]: 

=  vX  360.  CGS  units  (elst).      About  1.08  X  1013 13-033  4238 

=     3600.  coulombs 3-556  3025 

=        360.  CGS  units  (elmg) 2-556  3025 

1  absolute  unit  =  1  CGS  unit,  either  elst.  or  elmg 0-000  0000 

The  relations  to  other  measures  are  as  folio ws:f 
Coulombs  =  amperes  X  seconds. 

=  amperes  X  hours  X  3  600. 
=  volts  X  seconds  -4-  ohms. 
=  watts  X  seconds  -4-  volts. 
=  \/(  watts  X  seconds2  -*-  ohms). 
"  =  joules  -T-  volts. 

=  \/(  joules  X  seconds  -4-  ohms). 
41  =  farads  X  volts. 

=  microfarads  X  volts  -4- 1  000  000. 
Ampere-hours  =  coulombs  -f-  3  600. 
=  amperes  X  hours. 
=  volts  X  hours  -r-  ohms. 
=  watts  X  hours  -4-  volts. 
—  \/(  watts  X  hours2  -4-  ohms). 
=  joules  -*-  (volts  X  3  600). 
=  Vfjoules  X  hours  -4-  (ohms  X  3  600)]. 
Microcoulombs  =  microfarads  X  volts. 

t  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  peri- 
odically varying  quantities. 


ELECTRICAL   CAPACITY.  117 


ELECTRICAL     CAPACITY  [C,  c].      (Quantity  ^  electro- 
motive  force.) 

These  units  arte  used  to  measure  the  ability  of  a  body  (like  a  condenser) 
to  hold  charges  of  electricity  (measured  in  coulombs)  under  electrical 
stress,  pressure,  or  potential  (measured  in  volts).  The  capacity  of  a  con- 
denser is  greater  the  greater  the  charge  in  coulombs  that  it  will  hold  for  the 
same  pressure  in  volts,  or  the  less  the  pressure  in  volts  required  for  the 
same  charge.  The  capacity  in  farads  is  equal  to  the  charge  in  coulombs 
divided  by  the  electromotive  force  in  volts.  In  some  respects  an  electrical 
capacity  is  analogous  to  the  capacity  of  a  closed  vessel  to  hold  air  under 
pressure,  the  greater  the  pressure  the  greater  the  quantity  of  air,  yet  the 
capacity  of  the  vessel  remains  the  same  and  could  be  measured  in  terms 
of  the  air  and  its  pressure. 

The  unit  universally  used  is  the  microfarad  or  millionth  of  a  farad; 
by  farad  is  here  meant  the  international  farad  of  the  International 
Congress  of  1893  at  Chicago,  defined  as  equal  to  one  international  coulomb 
divided  by  one  international  volt.  It  was  made  legal  in  this  country  by 
Congress  in  1894,  and  is  accepted  by  the  National  Bureau  of  Standards. 
As  the  farad  is  an  inconveniently  large  unit,  never  occurring  in  practice, 
the  microfarad  is  generally  used.  The  electromagnetic  C.G.S.  unit  (or 
absolute  unit)  is  the  capacity  of  a  condenser  which  when  charged  at  one 
C.G.S.  unit  of  potential  will  hold  one  C.G.S.  unit  of  quantity.  The  elec- 
trostatic C.G.S.  unit  (or  absolute  unit)  is  similarly  defined  with  respect 
to  the  electrostatic  units  of  potential  and  quantity. 

By  "farad"  is  meant  the  international  farad,     v  is  the  velocity  of  light. 

Logarithm 

1  COS  unit  [elst]=  1  abstafarad 0-000  0000 

=  1015/v2  microfarad.    About  V»  X  10~5 6  045  7575 

=  109/?;2  farad.    About  V9  X  lO"11 12-045  7575 

=      l/v2  CGS  unit  (elmg).    About  V9  X  10"20 .    21-045  7575 

1  abstafarad  =>  1  CGS  unit  (elst) 0-000  0000 

I  microfarad  =v2X  1Q-15CGS  units  (elst).    About9X105 5.954  2425 

1  microcoulomb -«- 1  volt 0-000  0000 

=  0.000001  farad 6  000  0000 

=       10~15    CGS  unit  (elmg) 1 5  000  0000 

1  farad  [F]=  v2X  10~9  CGS  units  (elst).    About  9X  1011 11-954  2425 

=  1  000  000.  microfarads 6  000  0000 

1C-9  CGS  unit  (elmg) 9.QOO  0000 

1  international  farad  =  1  farad,  which  see  above 0-000  0000 

1  CGS  unit  (elmg)=    u2  CGS  units  (elst).     About  9X1020..  .   20-954  2425 

=  1015  microfarads 15  000  0000 

44  =  109  farads 90000000 

=      1  abfarad 0-000  0000 

1  abfarad  =  1  CGS  unit  (elmg). 0-000  0000 

1  absolute  unit  =  1  CGS  unit,  either  elmg  or  elst 0-000  0000 

The  relations  to  other  measures  are  as  follows:  t 
Farads  =  coulombs  -T-  volts. 

"       =  joules  of  stored  energy  X  2  rf-  volts2. 
Microfarads  =  coulombs  X  1  000  000.  -*•  volts. 
=  microcoulombs  -J-  volts. 
=  joules  of  stored  energy  X  2  000  000.  -f-  volts2. 

For  alternating -current  circuits -f 
Farads  =  amperes -5- (volts  X frequency X 6.283  19  t ). 

"        =  —  1  -v-  (ohms  reactance  X  frequency  X  6.283  19  J  )• 
Microfarads  =  amperes  X  1  000  000  -*- (volts X frequency X 6. 283  19  t). 
=  1  000  000. 4-  (ohms  reactance  X  frequency  X  6.283  19  j ). 

t  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  period- 
ically varying  quantities. 

t  Or  2n.     Aprx.  ^X 10,     Log  0-798  1799. 


118  INDUCTANCE. 


INDUCTANCE  [L,  1];  COEFFICIENT  of  SELF-  or 
MUTUAL  INDUCTION.  (E,m.f.  -f-  (current  -f  time)  ;  re- 
sistance X  time;  number  of  turnsXflux-7-currentj  kinetic 
energy -f- square  of  current.) 

These  units  are  used  to  measure  the  intensity  of  that  property  by  virtue 
of  which  an  electromotive  force  is  produced  by  changes  of  current  in  a 
neighboring  circuit  (as  in  a  transformer)  or  by  changes  of  current  in  the 
circuit  itself;  in  the  former  case  the  phenomenon  is  called  mutual  induc- 
tion, and  in  the  latter  self-induction.  An  electric  current  has  a  property 
analogous  in  some  respects  to  the  inertia  of  a  heavy  moving  body;  a  cur- 
rent resists  momentarily  any  change  in  its  strength,  just  as  a  heavy  moving 
body  resists  any  change  in  its  velocity.  When  a  current  is  started,  it  en- 
counters for  a  short  time  (see  time-constant  below)  a  counter-electromotive 
force  in  its  own  circuit  due  either  to  itself  (self-induction)  or  to  a  current 
in  the  opposite  direction  which  it  is  inducing  in  a  neighboring  circuit  (mutual 
induction).  Similarly,  if  stopped  it  tends  to  prolong  itself  either  in  its  own 
circuit  or  (by  induction)  in  a  neighboring  circuit;  the  quicker  the  change 
the  greater  this  tendency.  This  property  is  due  to  the  inductance. 

The  term  induction  in  electrodynamics  applies  broadly  to  the  general 
phenomenon  of  the  generation  of  an  electromotive  force  (which  may  or  may 
not  produce  a  current)  by  magnetic  flux,  whether  it  be  that  of  a  magnet  or 
that  surrounding  a  current.  The  terms  self  and  mutual  induction  apply 
to  the  special  cases  of  this  phenomenon  when  the  induction  is  produced  by 
a  current  in  its  own  circuit  or  in  a  neighboring  circuit  respectively,  there 
being  no  mechanical  motion.  The  terms  coefficient  of  self  or  coefficient 
of  mutual  induction  apply  to  the  numerical  value  of  the  intensity  of  this 
phenomenon,  by  which  it  is  measured;  the  terms  self-inductance  and 
mutual  inductance  are  now  more  generally  used  instead.  The  term 
inductance  applies  to  both  of  these  coefficients,  and  is  therefore  the 
measure  of  self  or  mutual  induction.  The  inductance  factor  is  the  ratio 
of  the  wattless  volt  amperes  to  the  total  volt  amperes;  or  the  ratio  of  the 
wattless  component  of  the  current  or  e.m.f .  to  the  total  current  or  e.m.f. 

The  inductance  depends  for  its  value  on  the  geometric  conditions  of  the 
circuit,  and  varies  with  the  size  and  shape  of  the  circuit  or  of  a  coil,  with 
the  square  of  the  number  of  turns  of  a  coil,  with  the  distance  between  the 
wires,  etc.;  it  also  varies  according  to  an  irregular  law  with  the  presence  of 
iron  or  other  magnetic  material.  An  inductance,  in  henrys,  is  measured 
by  and  is  equal  to  the  electromotive  force  induced,  in  volts,  divided  by  the 
rate  of  change  of  current  in  amperes  per  second/which  causes  it;  it  is  also 
equal  in  henrys  to  twice  the  kinetic  (magnetic)  energy  of  the  circuit  in 
joules  divided  by  the  square  of  the  final  current  in  amperes;  a  coefficient 
of  self-induction  in  henrys  is  also  equal  to  the  resistance  of  the  circuit  in 
ohms  multiplied  by  its  time-constant  (see  below)  in  seconds.  The  induct- 
ance of  any  particular  circuit  is  also  the  constant  relation  of  the  product 
of  the  magnetic  flux  and  th3  turns,  to  the  current  producing  the  flux. 

In  the  C.G.S.  system  of  units,  on  which  the  practical  unit  is  based,  this 
measure  happens  to  be  of  the  same  kind  as  a  length;  this  is  a  consequence 
of  what  is  called  a  "suppressed  factor"  in  the  dimensional  formula,  and  as 
it  is  misleading  and  answers  no  useful  purpose,  the  coincidence  should  not 
be  given  any  importance.  While  it  is  not  incorrect  to  express  inductances 
in  centimeters,  it  misleads  and  is  not  good  practice.  The  electromotive 
force  produced  by  inductance  is  generated  precisely  as  in  dynamos  by  the 
cutting  of  magnetic  flux  or  lines  of  force  and  at  the  rate  of  10s  such  lines 
(maxwells)  per  second  for  each  volt,  the  magnetic  flux  in  inductance  being 
that  produced  by  the  current,  and  is  proportional  in  amount  to  the  change 
in  the  current  strength.  In  the  case  of  dynamos  the  wire  moves  and  the 
flux  is  at  rest,  while  in  inductance  the  wire  is  at  rest  and  the  flux  moves. 

The  unit  now  most  generally  used  is  the  henry.  By  henry  is  here  meant 
that  of  the  International  Congress  of  1893  at  Chicago,  defined  as  the  induc- 
a  circuit  when  the  electromotive  force  induced  iu  this  circuit  ia  one 


INDUCTANCE.  119 

international  volt,  while  the  inducing  current  varies  at  the  rate  of  .one  inter- 
national ampere  per  second.  It  was  made  legal  in  this  country  by  Con- 
gress in  1894,  and  is  accepted  by  the  National  Bureau  of  Standards.  In 
some  of  the  relations  in  these  tables,  this  value  is  assumed  to  be  equal  to 
109  C.G.S.  electromagnetic  units  of  inductance.  This  unit  was  formerly,  and 
for  some  years  officially,  called  a  "quadrant."  If  the  self-inductance  of 
a  given  circuit  (usually  a  coil)  is  one  henry,  it  means  that  the  magnetic  lines 
of  force  corresponding  in  that  circuit  to  one  ampere  will  form  108  linkages 
of  unit  magnetic  lines  of  force  (maxwells)  with  that  circuit.  The  unit 
called  sec-ohm  was  at  one  time  used,  as  in  self-induction  the  inductance 
is  equal  to  the  product  of  the  resistance  in  ohms,  and  the  time-constant  of 
the  circuit  in  seconds. 

The  electromagnetic  C.G.S,  unit  (or  abs9lute  unit)  is  that  induction 
which  will  induce  one  C.G.S.  unit  of  electromotive  force  by  a  change  of  cur- 
rent at  the  rate  of  one  C.G.S.  unit  of  current  per  second.  It  happens  to  be 
numerically  equal  to  1  centimeter  of  length  and  is  sometimes  so  represented. 
The  electrostatic  C.G.S.  unit  (or  absolute  unit)  is  similarly  denned  with 
respect  to  the  electrostatic  units  of  electromotive  force  and  current. 

INDUCTANCE. 

v  is  the  velocity  of  light.  Logarithm 

1  CGS  unit  (elmg)  =         1  centimeter 0  000  0000 

=  0.001  microhenry 3-000  0000 

=  10-fJ  henry g.QOO  0000 

=  l/v2  CGS  unit  (elst)..     About  %  X  10" w  .  .  21-045  7575 

1  centimeter  inductance  =  1  CGS  unit  (elmg) 0  000  0000 

1  microhenry  =         1  000.  CGS  units  (elmg) 3-000  0000 

=  0.000  001   henry 6  000  0000 

1  millihenry  =  0.001  henry 3-000  0000 

1  henry  [H]=         10''  CGS  units  (elmg) 9-000  0000 

=  10  000.  kilometers,  or  1  earth's  quadrant 4  000  0000 

=  109A2  CGS  unit  (elst).    About  %  X  10-" 12-045  7575 

1  quadrant  =  1  henry 0-000  0000 

1  quad  =  1  quadrant  or  henry 0-000  0000 

1  sec-ohm  =  1  henry 0-000  0000 

1  CGS  unit  (elst)  =              v2  CGS  units  (elmg).    AboutQXlO2"  20-9542425 

=              r2  centimeters.     About  9  X  1020  .  .  .  .  20-9542425 

=  v2X  10~9  henrys.    About9X10n 11-9542425 

The  relations  to  other  measures  are  as  folio  ws:f 
Henrys  =  induced  volts-;- rate  of  change  of  amperes  per  second. 
=  time  constant  in  seconds  X  ohms. 

=  time  constant  in  seconds  X  applied  volts  -f-  final  amperes. 
=  joules  of  kinetic  energy  of  the  current  X  2  -r-  final  amperes2. 
=  joules  of  kinetic  energy  of  the  current  X  ohms2  X  2  •*•  applied  volts2. 

When  the  flux  is  due  only  to  the  current,  as  in  self-induction,  and  when 
there  is  no  magnetic  leakage: 
Henrys  of  self-induction: 

=  (maxwells  X  number  of  turns2)  -r-  (ampere-turns  X  108). 
=  (maxwells  X  number  of  turns)  -r-(  final  amperes  X  108). 
=  (number  of  turns  X  0.000  1)2X  1.256  64  }  -s-  oersteds. 

When  there  is  magnetic  leakage,  substitute  in  the  above  for  the  quantity 
" maxwells  X  number  of  turns,"  the  mean   flux  turns,  that  is,  the  "  mean 
maxwells X number  of  turns."     Thus: 
Henrys  of  self-induction: 

=  (mean  max  wells  X  number  of  turns)  -r-(  final  amperes  X  10s). 
When  the  flux  is  from  an  external  source  and  independent  of  the  current, 
as  in  mutual  induction,  and  when  there  is  magnetic  leakage: 
Henrys  of  mutual  induction: 

=  (mean  maxwells  through  the  secondary X secondary  turns) -s- (final 
amperes  in  primary  X  10s). 

t  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  period- 
ically varying  quantities. 

t  Or  4 rr+10.     Aprx.  1%.     Log  0  099  2099- 


120  INDUCTANCE;  TIME-CONSTANT. 

For  alternating  current  circuits  :f 

Henrys  =  induced  volts -s- (amperes X frequency  X 6. 283  19 J). 
=  ohms  reactance  -r-  (frequency  X  6. 283  19J). 

=  v/(ohms  impedance2  — ohms  resistance2)  -s-  frequency  X  6. 283  19  J. 
44        =vT( applied    volts  -r-  amperes)2  —  ohms     resistance2]  -5-  frequency  X 

6.283  19 1 
Inductance  factor: 

=  wattless  component  of  current  or  e.m.f.  -r- total  current  or  e.m.f. 
=  V(1—  power  factor2). 


TIME-CONSTANT    (of  inductive  circuits).      (Inductances 
resistance ;   time.) 

When  a  current  is  started  in  a  circuit  containing  inductance  of  the  kind 
called  self-induction,  as  is  the  case  for  instance  in  coils,  particularly  if  they 
have  many  turns  and  iron  cores,  the  current  will  not  reach  its  full  value 
Instantly,  but  owing  to  the  self-induction  it  will  at  first  be  opposed  by  a 
jounter-electromotive  force  due  to  the  inductance;  this  opposition  will  grow 
less  and  less  until  the  current  has  reached  its  full  strength.  It  is  similar  to 
what  takes  place  when  a  heavy  weight  like  a  street  car  is  started  to  move; 
its  inertia  will  at  first  oppose  the  moving  force,  but  this  opposition  will  grow 
less  and  less  as  the  speed  increases  until  the  full,  constant  speed  is  attained, 
and  the  inertia  will  then  offer  no  further  opposition. 

It  is  sometimes  of  importance  to  know  how  long  it  takes  before  the  cur- 
rent has  reached  its  ultimate  value,  but  theoretically  it  takes  an  infinite 
time,  and  therefore  it  is  usual  to  state  the  time  that  it  takes  the  current  to 
rise  to  a  certain  definite  fractional  part  of  its  full  value,  namely  nearly  % 
(the  exact  figure  is  given  below),  and  this  time  is  called  the  "time-constant" 
of  that  circuit.  This  time  in  seconds  (often  a  very  small  fraction  of  a 
second)  is  equal  to  the  self-induction  in  henrys  divided  by  the  resistance 
in  ohms;  or  instead  of  the  ohms  one  may  of  course  use  the  applied  volts 
divided  by  the  final  steady  current  in  amperes.  This  time-constant  is 
therefore  greater  the  greater  the  self-induction  and  the  less  the  resistance. 
It  gives  more  information  about  a  circuit  than  the  mere  inductance  does, 
as  it  includes  the  resistance;  the  self-inductance  of  a  coil,  for  instance,  is 
the  same  whether  the  wire  is  made  of  copper  or  of  a  high  resisting  metal, 
but  the  time-constant  is  less  in  the  latter  case. 

The  exact  fractional  part  of  the  full  value  of  the  current,  above  referred 
to,  is  (e—  l)-=-e,  in  which  e  is  the  base  of  the  Naperian  logarithms.  Numer- 
ically this  is  equal  to  63.212%,  or  nearly  %. 

The  unit  in  which  time-constants  are  always  given  is  the  second,  hence 
there  are  no  reduction  factors.  The  most  important  relations  to  other 
measures  are  as  follows: 

Time-constant  in  seconds  =  henrys -f- ohms  resistance. 

=  henrys  X  final  amperes -r- applied  volts. 

For  further  relations  see  those  for  henrys  under  inductance,  and  divide 
them  by  the  ohms  resistance. 


t  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  period- 
ically varying  quantities. 

I  Or  2*.     Aprx.  %  X  10.     Log  0-798  1799- 


FREQUENCY.  121 


FREQUENCY;    PERIODICITY;    PERIOD;    ALTERNA- 
TIONS.    (1-i-time;   time.) 

Frequency  or  periodicity  is  the  number  of  recurrences  or  cycles  of 
some  periodic  or  wave  phenomenon  or  oscillation  during  a  given  time  which 
is  always  understood  to  be  a  second  unless  otherwise  stated;  the  frequency 
always  refers  to  the  number  of  complete  cycles.  The  number  of  alterna- 
tions, however,  refers  to  the  number  of  changes  of  the  direction  or  to  the 
reversals,  and  therefore  refers  to  half- waves,  and  is  always  equal  to  double 
the  frequency,  if  the  time  is  the  same.  The  period  is  the  time  of  one  com- 
plete wave  or  oscillation  and  is  therefore  the  reciprocal  of  the  frequency. 
The  term  "  frequency"  is  the  one  most  generally  used,  and  always  refers 
to  a  second;  the  term  "  number  of  alterations"  is  preferred  by  some,  and 
when  it  refers  to  electric  currents  the  time  is  usually  a  minute;  the  term 
"period  "is  used  comparatively  rarely  as  a  measure,  its  use  being  gen- 
erally limited  to  scientific  discussions;  the  unit  is  generally  the  second. 

In  mathematical  discussions  of  electric  alternating-current  problems  the 
frequency  is  often  replaced  by  an  angular  velocity,  generally  represented 
by  a)  and  measured  in  terms  of  radians  per  second  (see  under  Angular  Veloc- 
ities above).  Then  w  =  2,7in,  in  which  a>  is  in  radians  per  second  and  n  is  the 
true  frequency  in  cycles  per  eecond,  a  cycle  being  here  considered  the 
same  thing  as  a  complete  revolution. 

The  frequency  is  also  equal  to  the  velocity  of  propagation  divided  by 
the  wave-length.  The  wave-lengths  are  therefore  measured  in  units  of 
length,  but  when  the  velocity  for  a  class  of  waves  is  a  constant  (as  those 
of  light  or  the  electromagnetic  waves),  the  wave-lengths  may  also  be  in- 
dicated in  units  of  time,  in  which  case  a  wave-length  becomes  equal  to  the 
period  of  the  wave.  Wave-length  should  not  be  confounded  with  the 
amplitude,  which  measures  the  intensity  of  the  wave  and  has  nothing 
to  do  with  the  frequency,  period,  or  wave-length. 

If  n  is  the  frequency  per  second  [CO],  then: 
the  period  in  seconds  =  1/n; 
the  number  of  alternations  per  minute  =  120n. 

If  n  is  the  number  of  alternations  per  minute,  then: 
the  frequency  per  second  =  n/120; 
the  period  in  seconds  =  120/n. 

If  n  is  the  period  in  seconds,  then: 

the  frequency  per  second  =  1/n; 

the  number  of  alternations  per  minute  =  120n. 

If  n  is  the  frequency  in  cycles  per  second,  and  (o  the  angular  velocity 
in  radians  per  second,  and  if  a  cycle  is  represented  by  one  complete  revolu- 
tion, then: 

a)  =  2i:n\  or 

n  =  0.159  155w;  and 
<u  =  6.283  19n. 
An  electrical  degree  is  the  360th  part  of  one  complete  cycle. 

For  the  relations  of  frequency  to  other  measures  see  those  relations  b^- 
tween  other  measures  which  involve  the  frequency,  chiefly  under  farads 
and  henrys. 


122  ELECTRICAL    ENERGY. 


KINETIC  ENERGY  of  a  CURRENT  in  a  CIRCUIT  [W]. 
(Inductance X current2;  power X time-constant  j  energy.) 

As  was  explained  above  under  "Time-constant,"  it  takes  an  appreciable 
time  to  start  a  current  in  a  circuit  having  inductance.  During  this  interval 
of  time  energy  is  being  stored  up  in  the  circuit,  which  is  given  back  again, 
usually  in  the  form  of  a  spark,  when  the  current  is  stopped;  this  is  called 
the  kinetic  energy  of  the  current  in  the  circuit.  It  has  an  analogy  in  the 
storing  of  energy  in  a  heavy  body  like  a  street  car  when  it  is  started  to  move, 
which  energy  is  given  back  again  when  the  body  is  stopped. 

This  kinetic  energy  is  greater  the  greater  the  self-induction  and  the 
greater  the  current.  It  is  equal  in  joules  to  half  the  product  of  the  self- 
indu'tance  in  henrys  and  the  square  of  the  current  in  amperes  at  any 
instant,  and  therefore  also  of  the  amperes  of  the  final  steady  current.  Sim- 
ilarly in  mechanics,  this  kinetic  energy  is  equal  to  half  the  product  of  the 
mass  and  the  square  of  the  velocity  at  any  instant,  and  therefore  also  of 
the  final  steady  velocity.  This  energy  is  stored  in  the  form  of  magnetic 
energy,  and  during  that  time  it  is  potential  energy;  it  remains  stored  as  long 
as  the  current  continues,  and  is  given  out  again  when  the  current  ceases 
In  a  transformer  it  is  the  kinetic  energy  of  the  primary  circuit  which  is  trans- 
mitted to  and  is  led  out  by  the  secondary.  The  practical  unit  is  the  joule, 
and  the  C.G.S.  unit  is  the  erg. 

The  chief  relations  -to  other  measures  are  as  follows- 

Joules  of  kinetic  energy  of  the  current  =  henrys  X  final  amperes2-?- 2. 
Ergs  of  kinetic  energy  of  the  current  =  henrys  X  final  amperes2  X  5  X  106. 

For  further  relations  see  under  units  of  Electrical  Energy,  below. 

ELECTRICAL  ENERGY  OR  WORK  [W].  (Quantity  X 
electromotive  force;  current  2X  resistance  X  time  j  power 
X  time  ;  power  -f-  frequency,) 

The  units  of  electrical  energy  are  all  convertible  directly  into  units  of 
other  kinds  of  energy,  as  energy  is  the  one  quantity  common  to  all  the 
systems  of  units.  The  electrical  and  absolute  units  have  therefore  been 
included  with  the  mechanical,  thermal,  and  other  units  in  the  general  table 
of  all  the  units  of  Energy  (r>.  74);  the  present  table  is  limited  to  a  few 
specific  values  and  to  some  relations  between  the  electrical  unit  of  energy 
and  other  electrical  units. 

The  energy  in  joules  delivered  to  a  circuit  is  equal  to  the  electromotive 
force  in  volts  multiplied  either  by  the  quantity  of  electricity  in  coulombs, 
or  by  the  product  of  the  current  in  amperes  and  the  time  in  seconds.  These 
apply  to  direct  currents;  in  alternating-current  calculations  involving 
energy,  the  power  and  not  the  work  done  is  generally  the  important  con- 
sideration; the  energy  of  alternating  currents  of  any  wave  form  delivered 
per  cycle,  in  joules,  is  equal  to  the  power  in  mean  watts  divided  by  the 
frequency.  For  the  energy  stored  in  a  current  see  the  preceding  section 
on  Kinetic  Energy. 

The  unit  universally  used  is  the  joule,  by  which  is  here  meant  the  joule 
of  the  International  Congress  of  1893  in  Chicago,  defined  as  equal  to  107 
C.G.S.  units  of  work  (ergs)  and  represented  sufficiently  well  for  practical 
use  by  the  energy  expended  in  one  second  by  an  international  ampere  in 
an  international  ohm ;  in  the  relations  in  these  tables  this  defined  value  is 
used.  It  was  made  legal  in  this  country  by  Congress  in  1894  and  is  accepted 
by  the  National  Bureau  of  Standards.  Sometimes  the  ampere-hour  is  used 
as  the  unit  of  electrical  quantity,  in  which  case  the  corresponding  unit  of 
energy  becomes  the  volt-ampere-hour,  usually  called  the  watt-hour;  the 
kilowatt-hour  (  =  1000.  watt-hours)  is  also  common.  The  electro- 
magnetic C.G.S.  unit  (or  absolute  unit)  is  the  erg,  defined  as  the  work 
of  one  dyne  acting  through  one  centimeter.  The  electrostatic  C.G.S.  unit 
(or  absolute  unit)  is  this  same  erg. 


ELECTRICAL   ENERGY.  123 


ELECTRICAL  ENERGY. 

Aprx.  means  within  2%. 

Logarithm 

1  CGS  unit  (elmg)  =  1  erg 0-000  0000 

1  CGS  unit  (elst)  =  1  erg 0-000  0000 

1  absolute  unit  =  1  erg 0  000  0000 

1  erg  =        1  CGS  unit  (elmg) 0-000  0000 

4     =        1  CGS  unit  (elst) 0-000  0000 

' '     =  lO-7  joule      7.000  0000 

1  microioule  =  10.  ergs 1-000  0000 

1  joule  [J]  =       10  000  000.  ergs y.QOO  0000 

=  0.000  277  778  watt-hour.    Aprx.  3/u •-*- 1  000 4-443  6975 

1  kilo.joule  =  1  000.  joules 3. 000  0000 

1  watt-hour  =  3  600.  joules 3-556  3025 

=      3.6  kilojoules 0-556  3025 

1  kilowatt-hour  =  3  600  000.  joules 6-556  3025 

=         1  000.  watt-hours 3-000  0000 

For  further  conversion  factors,  see  table  of  units  of  Energy,  page  74. 

The  relations  to  other  measures  are  as  follows:  f 
Joules  =  volts  X  coulombs. 

=  volts  X  amperes  X  seconds. 
=  volts2  X  seconds  -4-  ohms. 
=  amperes2  X  ohms  X  seconds. 
=  ohms  X  coulombs2  -f-  seconds. 
"        =  watts  X  seconds. 
Joules  of  stored  energy  =  farads  X  volts2  -f-  2. 

-microfarads  X  volts2 -=-2  000  000. 
Ergs  of  stored  energy  =  microfarads  X  volts2  X  5. 
Joules  of  kinetic  energy  of  the  current: 
=  henrys  X  final  amperes2 -4- 2. 
=  henrys  X  applied  volts2  -4-  (ohms2  X  2). 
=  time-constant   in   seconds  X  ohms  X  final  amperes2 -4- 2. 
=  time-constant  in  seconds  X  final  amperes  X  applied  volts -f- 2. 
=  time-constant  in  seconds  X  applied  volts2  -4-  (ohms  X  2). 
Ergs  of  kinetic  energy  of  the  current  =  henrys X final  amperes2X5X  106. 

When  the  flux  is  due  only  to  the  current,  as  in  self-induction,  and  when 
there  is  no  magnetic  leakage: 

Joules  of  stored  energy  =  max  wells  X  ampere-turns  -4-  (2  X  108). 
Ergg  of  stored  energy  =  maxwells  X  ampere-turns  -4-20. 

When  there  is  magnetic  leakage: 

Joules   of  stored  energy  =  mean  maxwells  X  ampere-turns  -4-2  X  10s. 
Erg.s  of  stored  energy  =  mean  max  wells  X  ampere-turns  -4-  20. 

When  the  flux  is  from  an  external  source,  and  independent  of  the  cur- 
.  rent,  as  in  mutual  induction,  and  when  there  is  no  magnetic  leakage: 
Joules  of  stored  energy  =  maxwells  X  ampere-turns -4-  10s. 
Ergs  of  stored  energy  =  maxwells  X  ampere-turns  -4- 10. 

When    there    is    magnetic    leakage    substitute    "mean   maxwells"    for 
"  maxwells." 

For  alternating  current  circuits:  t 
Joules  per  cycle  =  watts  -4- frequency. 

=  effective  amperes  X  effective  volts  X  cos  $-4-  frequency. 


t  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  peri- 
odically varying  quantities. 


124 


ELECTRICAL   POWER. 


ELECTRICAL     POWER    [P].      (Current  X  electromotive 
force ;  energy  -~  time  j     energy  X  frequency.) 

The  units  of  electrical  power  are  all  convertible  directly  into  units  cf 
other  kinds  of  power,  and  the  electrical  and  absolute  units  have  there- 
fore been  included  with  the  mechanical,  thermal,  and  other  units  in  the 
general  table  of  all  the  units  of  Power  (p.  80).  The  present  table  is  lim- 
ited to  a  few  specific  values  and  to  391116  relations  between  the  electrical 
unit  of  power  and  other  electrical  units. 

The  power  in  watts  is  equal  to  the  energy  in  joules  divided  by  the  time 
in  seconds;  this  applies  to  both  direct  and  alternating  currents  whether 
there  is  phase  shifting  in  the  latter  case  or  not.  According  to  Joule's 
law,  the  power  in  watts  is  also  equal  to  the  square  of  the  current  in  am- 
peres multiplied  by  the  resistance  in  ohms,  or  to  the  current  in  amperes 
multiplied  by  the  electromotive  force  in  volts,  or  to  the  square  of  the 
electromotive  force  in  volts  divided  by  the  resistance  in  ohms.  These 
apply  to  direct  currents;  they  apply  to  alternating  currents  also,  but 
only  under  certain  conditions,  the  chief  one  of  which  is  that  there  is  no 
reactance  in  the  circuit  and  therefore  no  phase  shifting  (caused  by  in- 
ductance or  capacity),  in  which  case  the  relations  refer  to  the  effective 
values  of  the  current  and  electromotive  force.  For  further  information 
treatises  on  alternating  currents  should  be  consulted.  In  any  case,  these 
relations  give  the  true  power  when  the  result  is  multiplied  by  the  power 
factor  (see  below). 

The  unit  universally  used  is  the  watt,  by  which  is  here  meant  the 
watt  of  the  International  Congress  of  1893,  in  Chicago,  defined  as  equal 
to  107  C.G.S.  units  of  power  (erg  per  second)  and  represented  sufficiently 
well  for  practical  use  by  the  work  done  at  the  rate  of  one  joule  per  second; 
in  the  relations  in  these  tables  this  defined  value  is  used.  It  was  made 
legal  in  this  country  by  Congress  in  1894,  and  is  accepted  by  the  National 
Bureau  of  Standards.  The  kilowatt  (  =  1000  watts)  is  also  common, 
The  electromagnetic  C.G.S.  unit  (or  absolute  unit)  is  an  erg  per 
second.  The  electrostatic  C.G.S.  unit  (or  absolute  unit)  is  the  same 
erjj  per  second. 

Power  Factor  is  a  term  used  to  show  the  amount  of  true  power  con- 
tained in  a  given  amount  of  apparent  power.  It  is  the  ratio  of  the  true 
power  to  the  apparent  power.  Its  use  is  limited  chiefly  to  electric  power 
generated  by  alternating  currents.  With  direct  electric  currents,  the 
power  is  equal  to  the  product  of  the  volts  and  the  amperes,  and  is  called 
watts;  with  alternating  currents,  however,  this  is  true  only  when  the 
volts  and  amperes  are  exactly  in  phase  with  each  other,  which  often  is  not 
the  case.  When  there  is  such  a  difference  in  phase,  that  is  when  the  current 
lags  behind  or  precedes  the  voltage,  their  product  is  only  apparent 
power  and  is  usually  measured  in  terms  of  the  product  of  the  volts  and 
the  amperes  and  called  volt-amperes.  If  the  true  power  in  such  a  casa 
is  measured  in  watts,  then  the  power  factor  will  be  the  number  of  watts 
divided  by  the  number  of  volt-amperes,  and  it  will  always  be  less  than 
unity,  in  practice  usually  between  07  and  0.95.  For  true  sine  waves, 
the  real  power  in  watts  is  equal  to  the  voltage  X  current  X  cos  0,  in  which 
<j>  is  the  angular  phase  difference;  hence  it  follows  that  in  such  cases  the 
power  factor  is  numerically  equal  to  cos  <f>,  whose  numerical  value  is  found 
directly  from  a  table  of  cosines.  Sometimes  the  power  factor  is  stated 
in  percent,  in  which  case  it  is  equal  to  the  above  figure  multiplied  by  100. 

The  inductance  factor  is  the  ratio  of  the  wattless  volt-amperes  to 
the  total  volt-amperes;  or  the  ratio  of  the  wattless  component  of  the 
current  or  the  e.m.f.  to  the  total  current  or  e.m.f.  The  sum  of  the  squares 
i)f  the  inductance  factor  and  the  power  factor  is  equal  to  unity . 


ELECTROCHEMICAL   EQUIVALENTS.  125 


.ELECTRICAL,   POWER. 

Logarithm 

1  CGS  unit  (elmg)  =  1  erg  per  second 0-000  0000 

1  CGS  unit  (elst)  =  1  erg  per  second 0-000  0000 

1  absolute  unit  =  1  erg  per  second 0  000  0000 

1  erg  per  second  ==        1  CGS  unit  (elmg) 0  000  0000 

=       1  CGS  unit  (elst) 0-000  0000 

=  10-7  watt 7.000  0000 

1  microwatt  =  10.  ergs  per  second 1  000  0000 

1  watt  [W,  w]  =  10  000  000.  ergs  per  second 7-000  0000 

1  joule  per  second 0  000  0000 

1  kilowatt  =  1  000  watts 3-000  0000 

For  further  conversion  factors,  see  table  of  units  of  Power,  page  80. 
The  relations  to  other  measures  are  as  follows:  f 
"Watts  =  volts  X  amperes. 
=  amperes2  X  ohms. 
=  volts2  -T-  ohms. 
=  coulombs  X  volts  -5-  seconds. 
=  coulombs2  X  ohms  -v-  seconds.2 
=  joules  -s-  seconds. 

For  alternating-current  circuits:  f 
Watts  =  volts  X  amperes  X  cos  <f>. 
=  amperes2  X  ohms  X  cos  0. 
=  volts2  X  cos  $  -H  ohms. 

=effec.  volts  Xeffec.  amp.  X  ohms  resistance -5- ohms  impedance. 
=  joules  per  cycle  X  frequency. 

Mean  watts  =  effective  volts  X  effective  amperes  X  cos  <£. 
Power  factor  =  true  power  in  watts  H- apparent  power  in  volt-amperes. 

=  energy  component  of  current  or  e.m.f. -f- total  current  or 

e.m.f. 

=  \/(  1  —  inductance  factor2 ) . 
Inductance  factor: 

=  wattless  component  of  current  or  e.m.f. -s- total  current  or  e.m.f. 
=  >/(!  —power  factor2). 


ELECTROCHEMICAL  EQUIVALENTS  and  DERIVA- 
TIVES. (Weight -7- quantity  of  electricity;  quantity  of 
electricity  -:-  weight ;  weight  -r-  energy  j  energy  -r-  weight.) 

The  electrochemical  equivalent  of  any  chemical  element  or  ion  is  the 
amount  by  weight  which  changes  its  chemical  combination  per  coulomb 
of  electricity  during  electrolysis.  For  these  and  their  derivatives  various 
compound  units  are  used  such  as  milligrams  per  coulomb,  grams  per 
ampere-hour,  pounds  per  ampere-hour,  etc.  The  relations  between 
most  of  these  are  simply  the  relations  between  the  respective  units  of 
weight,  which  see  under  Weights.  The  reciprocals  of  these  are  also  used 
frequently. 

The  electrochemical  equivalents  of  any  elements  or  ions  are,  according 
to  Faraday's  law,  proportional  to  their  atomic  weights  and  inversely 
proportional  to  their  changes  of  valency.  For  determining  the  actual 
values,  that  of  some  one  element  must  be  determined  experimentally, 
after  which  all  the  others  can  be  calculated.  In  the  following  relations 
the  value  taken  as  a  basis  is  the  electrochemical  equivalent  of  silver  adopted 
by  the  International  Electrical  Congress  of  Chicago  in  1893,  in  the  defi- 
nition of  the  ampere,  and  legal  in  this  country,  namely  0.001  118  gram 
per  coulomb.  The  atomic  weight  of  silver  used  in  these  relations  is  107.93, 

t  Treatises  on  alternating  currents  should  be  consulted  for  the  limiting 
conditions  under  which  these  relations  apply  to  alternating  or  other  peri- 
odically varying  quantities. 


126   ELECTROCHEMICAL  EQUIVALENTS. — DEPOSITS. 

which  is  the  usually  accepted  value  on  the  basis  of  0  =  16.  These  funda- 
mental values  correspond  with  the  usually  accepted  value  of  the  ionic 
charge,  96  540.  coulombs  per  monovalent  gram  ion,  within  the  limits  of 
accuracy  of  the  data.  For  a  complete  table  of  the  equivalents  and  their 
derivatives,  of  all  the  various  elements  and  for  various  changes  of  valency, 
accompanied  by  descriptions  of  how  to  use  them,  see  the  author's  Table 
of  Electrochemical  Equivalents  and  their  Derivatives,  in  Electrochemical  In- 
dustry, Jan.,  1903,  p.  169.  The  following  relations  apply  to  all  elements 
or  compounds.  (The  atomic  weights  are  all  based  on  oxygen  =  16,  but 
the  electrochemical  equivalents  are  independent  of  whether  the  atomic 
weights  are  based  on  0  =  16  or  on  H  =  l.) 

Milligrams  per  coulomb    =      0.010359X]        atomic  weight 
Grams  per  ampere-hour    =       0.037  291 X   \    ,  . weignt    . 

Pounds  per  ampere-hour  =  0.000  082  21 X  J  cnanSe  of  valency 

Grams  per  watt-hour  =0.037  291  X  1 

Kilograms  per  kilowatt  hour          =0.037  291  X  atomic  weight 

Kilograms  per  horse-power  hour  =  0.027  806  X    !* weignt 

Pounds  per  kilowatt-hour  =   0.082  21  X   I  cnange  of  val.  X  volts. 

Pounds  per  horse-power  hour          =  0.061  30  X  J 

Coulombs  per  milligram  =   96.54  X  1   chanee  of  valencv 
Ampere  hours  per  gram    =26.816  X  \  -^  F^. 

Ampere-hours  per  pound  =  12  164.  X  J      atomic  weight. 

Watt-hours  per  gram  =26.816X1 

Kilowatt-hours  per  kilogram        =26.816X       ohane-p  nf  val 
Kilowatt-hours  per  pound  =  12.164  X    [  -  ^  X  volts. 

Horse-power  hours  per  kilogram  =35.964  X        atomic  weight 
Horse-power  hours  per  pound        =16.313X  J 

Ionic  charge  for  a  monovalent  gram  ion  =  96  539.  coulombs. 

For  the  amount  of  gas  in  cubic  centimeters  at  0°  C.  and  760  mm  mer- 
cury pressure,  developed  at  one  electrode,  on  the  basis  that  one  gram  mole- 
cule of  a  gas  has  a  volume  of  22.38  liters,  the  following  relations  exist,  in 
which  n  is  the  number  of  atoms  per  molecule : 

Cb.  centimeters  per  ampere-hour  =834. 6-5-  (nX change  of  valency). 

Ampere-hours  per  cb.  centimeter  =  0.001 198  XnX  change  of  valency. 


ELECTROLYTIC  DEPOSITS.     (Mass -f- time;  mass  -5- 
surface.) 

Two  kinds  of  units  are  used  for  measuring  deposits,  such  as  those  in  elec- 
trolysis. One  is  for  measuring  the  weight  of  the  deposit  on  a  limited  sur- 
face, and  includes  such  units  as  grams  per  square  decimeter,  ounces 
per  square  foot,  etc.,  these  are  all  given  above  in  the  same  table  as  that 
for  Pressures  (page  63)  and  are  therefore  not  repeated  here 

The  other  kind  of  unit  is  for  measuring  the  rate  of  deposition,  and  in- 
cludes such  units  as  milligrams  per  second,  pounds  per  day,  tons 
per  year,  etc.;  the  reduction  factors  for  these  are  given  in  the  following 
table.  The  year  is  taken  as  equal  to  365M  days  and  the  ton  as  equal  to 
the  short  ton  of  2000.  pounds. 
Aprx.  means  within  2%. 

Logarithm 
1  pound  (av)  per  year  [Ib/yrJ: 

=   0.002  737  85  pound  per  day.    Aprx.  i^-f- 1000 3-4374098 

=  0.000  862  409  gram  per  minute.    Aprx.  %  •*- 1000 4-9357131 

I  kilogram  per  year  [kg/yr]: 

=  0.006  035  93  pound  per  day.    Aprx.  6 -f- 1  000 3-780  7440 

=  0.002  737  85  kilogram  per  day.    Aprx.  ^A^\  000 3-437  4098 

=  0.001  901  29  gram  per  minute.    Aprx.  19^-10  000 3-279  0473 


0.666  667  ounce  per  hour,  or? 

0.453  592  kilogram  per  day.    Aprx.  %  •*-  10 

0.314  995  gram  per  minute.    Aprx.  31  -*• 100 

0.182  625  ton  (short)  per  year.    Aprx.  H'e  •*•  10 

0.165  675  metric  ton  per  year.    Aprx. 


ELECTROLYTIC   DEPOSITS.  127 

Logarithm 

1  grain  per  hour  [g/h]: 

=  0.016  666  7  gram  per  minute,  or  Veo,  which  see  ..........   2-221  8487 

1  milligram  per  second  [mg/s]: 

=  0.06  gram  per  minute,  which  see  for  other  values  .........   2-778  1513 

1  pound  (av)  per  day  [Ib/day] 

=          365.25  P9unds  per  year.    Aprx.  11AX  100  .......  ,  .  .  .   2-562  5902 

165.675  kilograms  per  year.    Aprx.  Ve  X  1  000  ........    2-219  2560 

"-823  9087 
-656  6658 
498  3033 
-261  5602 

-.--.-  .        ._  .-219  2560 

=  0.041  6667  P9und  per  hour,  orV24  .....................    2-619  7888 

=  0.018  899  7  kilogram  per  hour.    Aprx.  19  ^  1  000  .......  „    2-276  4548 

I  ounce  (av)  per  hour  [oz/h]: 

1.5  pounds  per  day,  or%  .......................   0-176  0913 

=  0.680  389  kilogram  per  day.    Aprx.  68-*-  100  ............    1-8327571 

=  0.47  2  492  gram  per  minute.    Aprx.  "/a  •*•  10  .............    1-6743948 

I  kilogram  per  day  [kg/day]: 

=        805.238  pounds  per  year.    Aprx.  800  ...............   2-905  9244 

=        365.250  kilograms  per  year.    Aprx.  l\i  X  100  .........   2-562  5902 

=     0.694  444  gram  per  minute.     Aprx.  Vio  ...............    1-841  6375 

=     0.402619  ton  (short)  per  year.    Aprx.  Mo  .............    I  604  8944 

=     0  365  250  metric  ton  per  year.    Aprx.  iff-flO  .........    1-562  5902 

=  0.091  859  3  pound  per  hour.    Aprx.  Vn  ................    2  963  1230 

=  0.041  6667  kilogram  per  hour,  or  1/24  ..................    2-619  7888 

I  giam  per  minute  [g/min]: 

=     1159.54  pounds  per  year.    Aprx.  %Xl  000  ...........   30642869 

=       525.96  kilograms  per  year.    Aprx.  Vi9  X  10  000  .......    2-720  9527 

=  60  grams  per  hour  ............................    1-778  1513 

=    16.6667   milligrams  per  second,  or  ioo/6  ............  ----    1-2218487 

=   3  174  66  pounds  per  day.     Aprx.  s'4'w  ................   0-501  6967 

=    1.440  00  kilograms  per  day.     Aprx.  1%  ................  0-158  3625 

=  0.579772  ton  (short)  per  year.     Aprx.  ty  ..............    1-763  2569 

=   0.525  96  metric  ton  per  year.    Aprx.  10/i9  .............    1-720  9527 

=  0.132277  pound  per  hour.    Aprx.  %  •*•  10  ..............    1-1214855 

=  0.06  kilogram  per  hour.  ........................   2-778  1513 

1  ton  (short)  per  year  [tn/yr]: 

=   5.475  70  pounds  per  day.    Aprx.  %  .  .  .  :  .............   0-738  4398 

=   2.483  74  kilograms  per  day.    Aprx.  1%  ................    0-395  1058 

=    1.724  82  grams  per  minute.    Aprx.  %  .................   0-236  7431 

=  0.228  154  pound  per  hour.    Aprx.  %-i-lO  ...........  .  .    1-358  2288 

=  0  103  489  kilogram  per  hour.    Aprx.  3>i-5-10  ............    1-0148944 

1  metric  ton  per  year  [t/yr], 

=   6.035  93  pounds  per  day.    Aprx.  6  ...............  ----   0-780  7440 

=   2.737  85  kilograms  per  day.    Aprx.  3/n  X  10  ...........   0-437  4098 

=    1.901  29  grams  per  minute.     Aprx  l%o  ...............   0  279  0473 

=  0.251  497  pound  per  hour.    Aprx.  M  ..................    I  400  5328 

=  0.114  077  kilogram  per  hour.     Aprx.  8/7  ................    1-057  1986 

1  pound  (av)  per  hour  [lb/h]; 

=       8  766.  pounds  per  year.    Aprx.  J^X  10  000  ...........    3-9428015 

=  3  976.19  kilograms  per  year.    Aprx.  4  000  .............    3-599  4672 

=  10.886  2  kilograms  per  day.     Aprx.  11  ................    1-036  8770 


. 

7.559  87   grams  per  minute.    Aprx.  ^X  10  .............  0-878  5145 

=       4.383  tons  (short)  per  year.    Aprx.  %  X  10  ..........  0-641  7715 

=  3.976  19  metric  tons  per  year.    Aprx.  4  ...............  0-599  4672 

1  kilogram  per  hour  [kg/h]: 

=  19  325.7   pounds  per  year.     Aprx.  19  000  ..............  4-286  1356 

=       8  766.  kilograms  per  year.    Aprx.  H  X  10  000  .........  3-942  8015 

=  52.910  9  pounds  per  day.    Aprx.  53  ...................  1-723  5454 

=  16.666  7   grams  per  minute,  or  i»%  ....................  1-221  8487 

=  9.662  86  tons  (short)  per  year.    Aprx.  2%  or  %i  X  100  ----  0-985  1056 

=       8.766  metric  tons  per  year.    Aprx.  7/sX  10  ...........  0-942  8015 

1  ounce  (av)  per  minute  [oz/min]: 

=  3^  pounds  per  hour,  which  see  for  other  values  ..........  0-574  0313 


128 


ELECTROCHEMICAL   ENERGY. 


ELECTROCHEMICAL   ENERGY. 

The  term  electrochemical  energy  is  used  to  refer  to  electrical  energy 
when  it  performs  chemical  work,  or  to  chemical  energy  when  it  is  set  free  as 
electrical  energy.  The  most  convenient  unit  for  expressing  electrochemical 
energy  is  the  joule,  as  the  calculations  are  then  the  simplest;  the  calorie 
is,  however,  the  one  generally  used  in 'tables  which  give  the  energy  of  com- 
bination of  chemical  compounds;  values  in  calories  or  any  other  units  of 
energy  are  readily  reduced  to  joules  with  the  aid  of  the  reduction  factors 
given  under  Energy,  page  74.  Care  must  be  taken  to  distinguish  between 
the  large  and  the  small  calorie,  a  distinction  which  is  often  not  made  in 
text-books  and  tables. 

There  is  a  very  simple  and  direct  way  of  calculating  how  many  volts 
will  be  required  to  decompose  a  chemical  compound  electrolytically, 
or  how  many  volts  will  be  generated  in  a  battery  in  which  chemical 
compounds  are  formed,  when  the  heats  or  energies  of  combination  of  the 
compounds  are  known.  It  is  sometimes  called  Thomson's  law,  and  can  be 
directly  deduced  from  Faraday's  law  and  the  principle  of  the  conservation 
of  energy.  It  is  important  to  remember,  however,  that  the  number  of 
volts  thus  calculated  is  limited  to  that  required  to  supply  the  necessary 
energy  of  decomposition,  or  that  generated  in  a  battery  by  the  formation 
of  compounds;  it  includes  nothing  more.  The  actual  voltage  involves 
some  further  correction  factors,  which,  though  generally  small  as  com- 
pared with  the  voltage  of  decomposition,  are  sometimes  of  importance. 
Among  these  correction  factors  is  the  voltage  required  to  overcome  the 
resistance  of  the  electrolyte;  this  depends  on  the  current  flowing  and  on 
the  resistance  of  the  electrolyte.  Another  correction  factor  is  the  Gibbs 
or  Helmhpltz  temperature  coefficient,  namely  the  rate  of  change  of  the 
voltage  with  the  temperature;  this  falls  out  when  there  is  no  such  change, 
or  in  practice  when  this  change  is  inappreciable.  Another  correction  fac- 
tor is  what  is  called  the  "over-voltage" ;  this  depends  on  the  fact  that  it 
takes  different  voltages  to  set  free  the  same  gas  at  electrodes  of  different 
metals.  For  these  correction  factors  treatises  on  electrochemistry  should 
be  consulted. 

In  this  rule  for  calculating  the  voltage  due  to  the  heat  of  combination, 
which  is  given  below,  the  constant  is  based  on  the  relation  between  the 
joule  and  the  calorie  given  in  the  table  of  units  of  Energy,  and  on  the  same 
fundamental  electrochemical  and  chemical  constants  for  silver  which 
were  used  to  calculate  the  reduction  factors  for  Electrochemical  Equiva- 
lents above,  namely  0.001  118  gram  per  coulomb  as  the  electrochemical 
equivalent  of  silver,  and  107.93  as  the  atomic  weight  of  silver,  on  the  basis 
ofO  =  16.  These  fundamental  values  correspond  with  the  usually 
accepted  value  of  the  ionic  charge,  96  540.  coulombs  per  monovalent  gram 
ion,  within  the  limits  of  accuracy  of  the  data.  For  the  heats  of  com- 
bination of  compounds,  see  reference-books  on  this  subject;  the  data  are 
usually  given  in  calories  per  gram  molecule,  but  care  must  be  taken  to  find 
out  whether  the  large  calorie  or  the  small  calorie  is  the  one  meant;  also 
whether  it  is  a  gram  molecule  or  a  kilogram  molecule  that  is  meant ;  care 
must  also  be  taken  to  see  that  the  given  number  of  calories  apply  to  the 
exact  decomposition  or  combination  under  consideration,  whether  gases 
are  set  free  and  escape  as  such,  or  whether  they  recombine,  and  if  the 
latter,  whether  this  recombination  enters  into  the  electrochemical  reaction 
or  whether  it  is  purely  chemical,  as  by  local  action,  also  whether  water  or 
some  other  accompanying  product  is  formed  or  decomposed,  etc. 


ELECTROCHEMICAL  ENERGY. — RELUCTANCE.  129 


Rule  for  calculating  the  voltage  of  decomposition  or  composi- 
tion, from  the  heat  of  combination. 

Logarithm 

For  monovalent  ions,  or  for  one  equivalent  weight: 
The  number  of  volts: 

=  the  number  of  kilogram  calories  per  gram  molecule  X 

0.043  363  Aprx.  % -i- 10. ...  2.637  n62 

=  the  number  of  kilogram  calories  per  kilogram  molecule, 

or  gram  calories  per  gram  molecule,  X  0.000  043  363. 

Aprx.  %-MO  000 5.637  1162 

=  tne  number  of  kilogram  calories  per  gram  molecule  -r 

23.061.  Aprx.%XlO . 1-3628838 

=  the  number  of  kilogram  calories  per  kilogram  molecule, 

or  gram  calories  per  gram  molecule,  -^23  061.     Aprx. 

%  X  10  000 4.362  8838 

For  miiltivalent  ions  divide  the  volts  thus  obtained  by  the  valency. 

That  is,  for  bivalent  ions  calculate  the  voltage  from  the  above  and  then 
divide  by  2;  for  trivalent  ions  divide  by  3,  etc.  Or  the  number  of  caloriea 
may  be  reduced  to  that  for  one  equivalent  weight  (by  dividing  the  calories 
by  the  valency),  and  the  above  rules  will  then  give  the  volts  directly. 


MAGNETIC  RELUCTANCE  [61,  R];  MAGNETIC  RESIS- 
TANCE. (Magnetomotive  force -v- flux;  length -~ (surface 
X  permeability) .) 

Reluctance  measures  the  amount  by  which  the  material  in  a  magnetic 
circuit  or  part  of  a  circuit  resists  or  opposes  th^  flux;  the  more  it  resists, 
the  greater  the  reluctance ;  it  is  the  opposite  to  permeance  and  numerically 
it  is  equal  to  the  reciprocal  of  permeance.  As  the  reluctance  of  a  centi- 
meter cube  of  air  (or  more  correctly,  of  a  vacuum)  in  th3  C.  G.S  system  is,  by 
definition,  equal  to  unity  (that  is,  to  one  oersted),  it  follows  that  the  amount 
of  reluctance  of  any  magnetic  circuit  or  any  part  of  it  also  represents  the 
number  of  times  that  its  reluctance  is  greater  than  that  of  the  air  of  the  same 
volume  and  shape.  Reluctance  is  analogous  to  resistance  in  an  electric 
circuit,  but  differs  in  these  three  important  features:  (1)  in  circuits  con- 
taining magnetic  materials  it  generally  varies  very  greatly  with  the  flux 
density,  being  constant  only  for  air  or  other  diamagnetic  materials ;  (2)  there 
is  no  such  a  thing  as  a  magnetic  insulator,  that  is,  an  infinitely  great  reluct- 
ance; (3)  maintenance  of  a  flux  through  a  reluctance  does  not  necessarily 
require  the  continuous  expenditure  of  energy.  A  reluctance  in  oersteds 
is  equal  to  the  magnetomotive  force  in  gilberts  divided  by  the  flux  in  max- 
wells The  reluctance  of  a  circuit  in  oersteds  is  equal  to  the  length  of  the 
circuit  in  centimeters  divided  by  its  cross-section  in  square  centimeters  and 
then  either  multiplied  by  the  reluctivity  or  divided  by  the  permeability; 
but  in  magnetic  materials  the  reluctivity  and  the  permeability  vary  with 
the  density  of  the  flux  in  the  circuit,  hence  this  method  of  calculation  is 
practicable  only  for  non-magnetic  materials  like  air,  the  permeability  of 
which  is  constant  and  is  equal  to  unity;  it  may  be  applied  to  air-gaps,  for 
instance.  The  reluctance  is  not  used  very  often  in  magnetic  calculations, 
these  beino;  usually  based  on  the  property  called  permeability,  as  is  ex- 
plained bek)w  under  the  units  of  magnetizing  force. 

The  only  unit  used  is  the  C.G.S.  unit  called  an  oersted,  which  is  the 
reluctance  through  which  a  C.G.S.  unit  of  magnetomotive  force  (called  a 
gilbert)  will  produce  a  C.G.S.  unit  of  flux  (called  a  maxwell). 

1  CGS  unit  (elmg)  =  l  oersted. 
\  oersted  =  1  CGS  unit  (elmg). 


130  MAGNETIC   RELUCTIVITY. 

The  relations  to  other  measures  are  as  follows: 
Oersteds: 

=  gilberts  -r-  maxwells. 

= gilberts -5-  (gausses  Xsq.  centimeters  section). 
=  ampere-turns  X  1.256  64f  -=-  max  wells. 

=  ampere-turns  X  1.256  64f  •*•  (gausses  X  sq.  centimeters  section). 
=  CGS  unit  of  current-turns  X  12.566  4J  -^-maxwells 
=  CGS  unit  current-turns X  12. 566  4  J-r- (gausses Xsq.  cm.  section). 
=  1  -5-  permeance  in  CGS  units. 

=  centimeters  length -5- (sq.  centimeters  section  X  permeability). 
=  centimeters  length X reluctivity -r-sq.  centimeters  section. 
=  inches  length  X  0.393  700  •*-  (sq  inches  section  X  permeability). 
=  inches  length X reluctivity  X 0.393  700 -r-sq.  inches  section. 
=  (number  of  turns  X  0.000  1  )2  X  1 .256  64f  -5-  henrys. 
Oersteds  in  iron  =  oersteds  in  air -r- permeability, 
in  air  =  oersteds  in  iron  X  permeability 


MAGNETIC  RELUCTIVITY  [v];  SPECIFIC  MAGNET- 
IC RELUCTANCE;  MAGNETIC  RESISTIVITY; 
SPECIFIC  MAGNETIC  RESISTANCE.  (Imperme- 
ability j  magnetizing  force  -f-  magnetic  induction  j  reluc- 
tance X  surface  —length  ;  reluctance  -h  reluctance.) 

Reluctivity  measures  the  number  of  times  that  a  material  resists  or 
opposes  magnetic  flux,  more  than  air  (or  vacuum)  does.  It  is  a  property 
of  a  material  and  numerically  it  is  equal  to  tha  reciprocal  of  the  permea- 
bility, which  see.  It  corresponds  in  some  respects  to  resistivity  or  specific 
resistance  in  electrical  circuits.  It  is  specific  reluctance  based  on  air;  but 
as  the  reluctivity  of  air  is  unity,  that  of  any  other  material  is  numerically 
equal  to  the  reluctance  in  oersteds  of  a  centimeter  cube  of  that  material. 
Reluctivity  is  seldom  used  in  magnetic  calculation,  its  reciprocal,  the 
permeability,  being  used  instead.  Like  permeability,  it  is  a  property  of 
the  material  in  a  magnetic  circuit,  and  its  values  are  different  with  different 
flux  densities. 

There  are  no  units,  as  it  is  a  mere  ratio.  The  relations  of  reluctivity  to 
other  magnetic  measures  are  the  reciprocals  of  those  for  permeability. 

The  chief  relations  are: 
Reluctivity  =  1  •+•  permeability.' 

=  sq.  centimeters  section  X  oersteds -r- centimeters  length. 

=  sq.  inches  section X oersteds X 2. 540  01  ^-inches  length. 


MAGNETIC  PERMEANCE;  MAGNETIC  CONDUCT- 
ANCE j  MAGNETIC  CAPACITY.  (1 -reluctance  j 
flux  -f-  magnetomotive  force  j  surface  -~-  (length  X  permea- 
bility).) 

Permeance  measures  the  amount  by  which  a  given  magnetic  circuit  con- 
ducts the  flux,  the  better  it  conducts,  the  greater  the  penmeance;  it  is  the 
opposite  to  reluctance,  and  numerically  it  is  equal  to  the  reciprocal  of  reluct- 
ance. It  is  analogous  to  conductance  in  an  electrical  circuit  Just  as  the 
reluctance  of  a  given  magnetic  circuit  refers  to  that  particular  circuit  or 
its  parts,  so  permeance  refers  to  a  particular  magnetic  circuit  or  its  parts. 
In  referring  to  the  property  of  the  material  itself,  independently  of  its 

t  Or  4;r/10.     Aprx.  Y*  X  10.     Log  0  099  2099- 
J  Or  4*.     Aprx.  Y%  X  100.     Log  1-099  2099 


PERMEANCE .  —PERMEABILITY.  131 

size  and  shape,  the  term  permeability  or  specific  permeance  is  used,  which 
see  below.  When  a  given  magnetic  circuit  contains  a  magnetic  material 
like  iron,  the  permeance  does  not  remain  constant  like  the  conductance  of 
an  electric  circuit,  but  it  varies  very  greatly  with  the  density  of  the  flux; 
it  is  constant  only  for  air.  Permeance  is  seldom  used  in  magnetic  calcu- 
lations (see  under  magnetizing  force).  It  can  always  be  avoided  by  using 
the  reciprocal  of  the  reluctance  instead ;  both  may  be  avoided  by  basing  the 
calculations  on  the  permeability. 

The  unit  is  the  C.G.S.  unit,  which  is  the  reciprocal  of  the  absolute  or 
C.G.S.  unit  of  reluctance,  that  is,  the  reciprocal  of  an  oersted;  it  has 
no  name.  The  permeance  of  a  centimeter  cube  of  air  (or  vacuum)  between 
two  parallel  sides  is  unity  in  the  C.G.S.  system.  Hence  the  permeance  of 
a  magnetic  circuit  or  of  a  part  of  it,  also  represents  the  number  of  times 
that  its  permeance  is  greater  than  that  of  an  equal  volume  of  air  of  the  same 
size  and  shape.  It  is  equal,  in  C.G.S.  units,  to  the  permeability  multiplied 
by  the  cross-section  in  square  centimeters  and  divided  by  the  length  in 
centimeters. 

The  relations  to  other  measures  are  as  follows: 
Permeance  =  maxwells  •*•  gilberts. 

=  1  -T-  oersteds. 

=  permeability  Xsq.  centimeters  sections  centimeters  length. 

=  permeability  X  sq.  inches  section  X  2.540  01  -r- inches  length. 
Permeance  of  iron  =  permeance  of  air  X  permeability. 


of  air    =  permeance  of  iron  -5- permeability. 
See  also  the  reciprocals  of  the  relations  given  under  Rel 


uctance. 


MAGNETIC  PERMEABILITY  [u] ;  SPECIFIC  PERI&E, 
ANCE  j  MAGNETIC  CONDUCTIVITY.  (Magnetic 
induction  -v-  magnetizing  force  \  flux  density  -r-  flux  density ; 
permeance  -j-  permeance  ;  1  -~  reluctivity.) 

Permeability  is  a  very  important  quantity  in  most  magnetic  calculations ; 
it  measures  the  number  of  times  that  a  material  conducts  or  aids  magnetic 
flux  better  than  air  does;  if,  for  instance,  the  permeability  of  a  certain 
kind  of  iron  under  certain  conditions  is  300  ,  it  means  that  it  conducts 
magnetic  flux  or  lines  of  force  300.  times  as  well  as  an  equal  amount  of  air 
would  under  the  same  conditions,  that  is,  it  is  300.  times  as  permeable  to 
the  flux  .  It  may  be  said  to  be  magnetic  conductivity  compared  to  air  as  a 
standard.  If  a  given  magnetomotive  force  produces  a  certain  amount  of 
flux  in  a  given  circuit  of  air,  it  will  produce  300.  times  this  flux  if  the  air 
were  replaced  by  this  kind  of  iron.  It  follows  from  this  that  the  magnet- 
izing force  (usually  represented  by  H)  multiplied  by  the  permeability  (n) 
gives  the  induction  (B)  in  the  iron,  that  is,  the  flux  density  in  the  iron; 
or  permeability  =  induction  -s- magnetizing  force;  this  is  further  explained 
under  the  units  of  magnetizing  force.  Permeability  is  an  inherent  property 
of  materials  and  its  values  are  usually  given  in  tables  or  curves.  It  is  also 
the  same  as  specific  permeance,  which  is  the  permeance  of  a  material  as 
compared  with  that  of  air;  as  the  permeance  of  a  centimeter  cube  of  air 
is  unity,  the  law  ,above  given  follows.  Permeability  is  the  reciprocal  of 
reluctivity.  It  is  analogous  to  conductivity  or  specific  conductance  in 
electrical  calculations,  with  a  very  important  difference,  however,  namely 
that  while  the  electrical  conductivity  of  a  material  is  constant  and  does  not 
change- with  the  current  which  flows,  the  permeability  of  the  chief  magnetic 
materials  varies  very  greatly  with  the  density  of  the  flux,  and  its  values 
must  therefore  be  given  for  each  flux  density,  for  which  reason  they  are 
usually  given  in  the  form  of  curves  called  permeability  curves  or  magneti- 
zation curves.  For  air,  however,  the  permeability  is  always  constant  an^ 
is  numerically  equal  to  unity.  Paramagnetic  bodies  are  those  whojH* 
permeability  is  greater  than  unity,  and  diamagnetic  bodies  those  whose 
permeability  is  less  than  unity. 


132     PERMEABILITY. — MAGNETOMOTIVE   FORCE. 

There  are  no  units  of  permeability,  as  it  is  a  mere  relation,  number,  or 
ratio  between  two  quantities  of  the  same  kind.  In  this  respect  it  is  like 
specific  gravity. 

For  a  brief  description  of  calculations  involving  permeabilities  see  under 
the  units  of  Magnetizing  Force. 

The  relations  to  other  measures  are  as  follows: 
Permeability: 

=  gausses  in  iron  -5-  gausses  in  air. 

=  inch  gausses  in  iron-:- inch  gausses  in  air. 

=  permeance  of  iron  -f-  permeance  of  air. 

=  oersteds  of  air  -J-  oersteds  of  iron. 

=  1-7-  reluctivity. 

=  1  +  [susceptibility  X  12.566  4  (or  4w)]. 

=  gausses -v- gilberts  per  centimeter. 

=  gausses  X  0.7  95  77  5  (or  10-^-4^) -J- ampere-turns  per  centimeter. 

=  gausses  X  2.540  01  -i- gilberts  per  inch. 

=  gausses  X  2. 021  27  -=-  ampere-turns  per  inch. 

=  inch  gausses  X  0.393  700 -h  gilberts  per  inch. 

=  inch  gausses  X  0.3 13  297  -f-  ampere-turns  per  inch. 

=  gausses  X  0.07  9  577  5  (or  1  -f-4«)-*-CGS  unit  current-turns  per  cm. 

=  centimeters  length -j-(sq.  centimeters  section X oersteds). 

=  inches  length  X  0.393  700-r-(sq.  inches  section  X  oersteds). 

=  permeance  X  centimeters  length  -j-sq.  centimeters. 

For  the  relations  with  maxwells  substitute  for  "gausses,"  in  any  of  the 
above,  "maxwells -j-sq.  centimeters  section";  and  for  "inch gausses,"  sub- 
stitute "max  wells -r-sq.  inches  section." 


MAGNETIC    SUSCEPTIBILITY  [>].      (Intensity   of   mag, 
netization  H-  magnetizing  force.) 

This  quantity  is  used  chiefly  in  physical  conceptions;  it  is  somewhat 
similar  to  permeability  in  that  it  expresses  the  magnetizability  of  a  sub- 
stance. It  is  equal  to  the  intensity  of  magnetization  (see  below)  divided 
by  the  magnetizing  force  which  produces  it.  There  are  no  units,  as  it  is 
a  mere  ratio  or  number.  Its  relation  to  permeability  is  as  follows: 
Susceptibility  =  (permeability  -1)X  0.07 9  577  5  (or  l-Mjr). 


MAGNETOMOTIVE  FORCE  [m.m.f.  SF,  F]j  AMPERE- 
TURNS  [a-t];  MAGNETIC  POTENTIAL;  DIFFER- 
ENCE OF  MAGNETIC  POTENTIAL;  MAGNETIC 
PRESSURE.  (Current  X  turns ;  flux X reluctance ;  energy  -~ 
pole  strength.) 

These  units  are  used  to  measure  the  magnetic  pressure  or  "motive  force" 
which  produces  or  ends  to  produce  a  magnetic  flux  in  a  magnetic  circuit, 
just  as  an  electromotive  force  tends  to  produce  a  current  of  electricity,  or 
a  pressure  of  water  tends  to  produce  a  flow  of  water  and  might  similarly 
be  called  the  hydraulic  motive  force.  In  practice  magnetomotive  forces  are 
generally  produced  by,  and  are  often  measured  in  terms  of,  what  are  called 
ampere-turns;  this  term  means  the  product  of  an  electric  current  in  am- 
peres and  the  number  of  turns  or  windings  of  the  coil  through  which  it 
flows;  such  a  current-carrying  coil  produces  a  definite  magnetomotive 
force,  which  in  turn  produces  an  amount  of  flux  dependent  on  the  amount 
of  reluctance  in  the  whole  magnetic  circuit.  According  to  the  laws  of 
electromagnetism,  the  magnetomotive  force  in  C.G.S.  units  (gilberts)  is 
always  numerically  equal  to  the  ampere-turns  multiplied  by  4^-7-10,  whether 


MAGNETOMOTIVE    FORCE.  133 

there  is  iron  in  the  magnetic  circuit  or  not.  The  magnetomotive  force  in 
gilberts  in  any  closed  loop  encircling  a  long  straight  wire  through  which  a 
current  passes  is  4n  times  the  C.G.S.  unit  of  current,  or  4?r-:-10  times  the 
number  of  amperes.  It  is  always  directly  proportional  to  the  current  pro- 
ducing it.  The  law  of  the  magnetic  circuit  is  similar  to  Ohm's  law  for  the 
electric  circuit,  namely  that  the  flux  (corresponding  to  the  current)  is  equal 
to  the  magnetomotive  force  (corresponding  to  the  electromotive  force), 
divided  by  the  reluctance  (corresponding  to  the  resistance);  hence  the 
magnetomotive  force  in  C.G.S.  units  (gilberts)  is  equal  to  the  flux  in  C.G.S. 
units  (maxwells)  multiplied  by  the  reluctance  in  C.G.S.  units  (oersteds). 
This  is  true  whether  there  is  iron  in  the  magnetic  circuit  or  not. 

Most  calculations  occurring  in  practice  are  simplified  by  using  the  quan- 
tity called  the  magnetizing  force  instead  of  the  magnetomotive  force,  as  is 
explained  below  under  the  units  of  Magnetizing  Force. 

There  are  three  units  of  magnetomotive  force  in  use;  the  more  general, 
and  often  the  more  convenient  one,  is  the  ampere-turn,  which  is  equal  to 
the  magnetomotive  force  produced  by  one  ampere  flowing  once  around  a 
magnetic  circuit;  this  is  irrespective  of  the  shape  or  size  of  the  electric  or 
magnetic  circuit;  the  latter  affect  only  the  reluctance  of  the  magnetic  cir- 
cuit and  thereby  the  resulting  flux.  This  unit  bears  an  incommensurate 
relation  to  the  absolute  magnetic  units  owing  to  the  factor  4?r. 

The  second  usual  unit  is  the  electromgaiietic  C.G.S.  unit  (or  absolute 
unit),  called  a  gilbert.  Its  definition  is  based  on  the  fact  that  the  magneto- 
motive force  produced  in  any  one  closed  loop  around  a  long  straight  wire 
through  which  a  C.G.S.  unit  of  current  (or  10  amperes)  flows,  is  4?r  C.G.S. 
units  of  magnetomotive  force;  hence  one  such  unit  is  equal  to  l-r-4;r  of 
this  magnetomotive  force.  It  may  also  be  defined  as  that  magnetomotive 
force  which  will  produce  a  flux  of  one  C.G.S.  unit  (maxwell)  through  one 
C.G.S.  unit  of  reluctance  (oersted). 

The  third  unit  is  like  the  ampere-turn  except  that  the  current  is  the 
C.G.S.  unit  of  current  (10  amperes)  instead  of  the  ampere.  This  unit  is 
therefore  equal  to  10  times  the  ampere-turn  unit.  It  has  no  name  other 
than  the  C.G.S.  unit  car  rent-turn.  It  is  never  used  in  practice. 

Logarithm 

1  CGS  unit  (elmg)  =  1  gilbert,  which  see  for  other  values. 
1  gilbert: 

1  CGS  unit  (elmg)  of  magnetomotive  force. 

=     0.795  775  (or  10/4?r)  ampere-turn.    Aprx.  subtr.  V5  .  .  .  .    1-900  7901 

=  0.079  577  5  (or  l/4;r)  CGS  unit  of  current-turn.    Aprx.8/ioo   2-900  7901 

1  ampere-turn  =  1.256  637  (or  4?r/10)  CGS  units.    Aprx.  add  Y±  0-099  2099 

=  1.256637  (or  4 7T/10)  gilberts.    Aprx.  add  M-  •    0-0992099 

0.1  CGS  unit  of  current-turn 1-000  0000 

1  CGS  unit  of  current-turn: 

=  12.566  37  (or  4*)  CGS  units  (elmg).     Aprx.  10% 1-099  2099 

=  12.566  37  (or  4;r)  gilberts.    Aprx.  10% 1-099  2099 

10.  ampere-turns 1-000  0000 

The  relations  to  other  measures  are  as  follows : 
Gilberts: 

==  ampere-turns  X  1.256  64. t 

=  CGS  unit  current-turns  X  12. 566  4.J 

=  maxwells  X  oersteds. 

=  max  wells  -^permeance  (in  CGS  (elmg)  units). 

=  maxwellsXcm  length -5- (sq.  cm  section  X  permeability). 

=  maxwells  X  inches  length  X  0.393  700 -H  (sq.   inches  section  X  permea- 
bility). 

=  maxwells  X  centimeters  length -=-sq.  centimeters  section.     For  air. 

=  maxwells  X ins.  length X 0.393  700-n sq.  ins.  section.     For  air. 

=  gausses  X  centimeters  length  -r-  permeability. 

=  gausses  X  inches  length  X  2  540  01  -f-  permeability. 

=  inch  gausses  X  inches  length  X  0.393  700  •*•  permeability. 

=  gausses  X  centimeters  length.     For  air. 

=  gausses  X  inches  length  X  2.540  01 .     For  air. 

=  inch  gausses  X  inches  length  X  0.393  700.     For  air. 

=  gausses  X  oersteds  X  sq.  centimeters  section. 

t  Or  47T/10.     Aprx.  add  M-     Log  0-099  2099- 
\  Or  4*.     Aprx.  y&X  100.     Log  1.099  2099- 


134  MAGNETOMOTIVE    FORCE. — MAGNETIZING    FORCE. 

Gilberts  for  iron  =  gilberts  for  air -5- permeability. 
Gilberts  for  air  =  gilberts  for  iron  X  permeability. 
Ampere-turns: 

=  gilbertsX 0.795  775.f 

=  COS  unit  current-turns  X  10, 

*=  max  wells  X  oersteds  X  0.795  775.  t 

=  maxwells  X  0.795  775f-*- permeance  (in  CGS  (elmg)  units). 

=  maxwells X cm  length XO. 795  775f-^(sq.  cm  section X permeability). 

=  maxwells  X  ins.  length  X  0.313  296-Ksq.  ins.  section  X  permeability), 

=  maxwells  X  cm  length  X  0.795  775f  •*-  sq.  cm  section.     For  air. 

=  maxwells X ins.  lengthXO.313  296-5-sq.  ins.  section.     For  air. 

=  gausses  X centimeters  length  X  0.795  77 5f-*- permeability. 

=  gausses  X  inches  length  X  2. 021  27  -5- permeability. 

=  inch  gausses X inches  lengthXO.313  296 -s- permeability. 

=  gausses X centimeters  length  X 0.795  775. t     For  air. 

=  gausses  X  inches  length  X  2. 021  27.     For  air. 

=  inch  gausses  X  inches  length  X  0.313  296.     For  air. 

=  gausses  X  oersteds  X  sq.  centimeters  section  X  0.795  775. f 
Ampere-turns  for  iron  =  ampere-turns  for  air -s- permeability. 
Ampere-turns  for  air  =  ampere-turns  for  iron X permeability. 
CGS  unit  of  current-turns  =  ampere-turns  X  0.1. 

=  gilberts  X  0.07  9  577  54 
(For  further  relations  divide  those  for  ampere-turns  by  10.) 

When  the  flux  is  due  only  to  the  current,  as  in  self-induction,  and  when 
there  is  no  magnetic  leakage: 
Ampere-turns  =ergs  of  kinetic  energy  of  the  current  X  20  -5-  max  wells. 

=  joules  of  kinetic  energy  of  the  current  X  2  X  lO8-?-  max  wells. 

=  maxwells  X  number  of  turns2  -T-  (henrys  X  108). 

When  the  flux  is  from  an  external  source,  and  independent  of  the  current 
as  in  mutual  induction,  and  when  there  is  no  magnetic  leakage: 
Ampere-turns  =ergs  of  kinetic  energy  of  the  current  X  10  -T-  maxwells. 

=  joules  of  kinetic  energy  of  the  current  X  10s -4- maxwells. 


MAGNETIZING  FORCE  [X,  H]J  MAGNETOMOTIVE 
FORCE  per  CENTIMETER j  MAGNETIC  FORCE; 
FIELD  INTENSITY.  (Turns  X  current  -f-  length  j  magneto- 
motive force -T- length  j  induction  -f-  per  me  ability  j  flux  den- 
sity -5- permeability  j  forces- pole  strength.) 

This  quantity,  which  is  one  of  the  most  important  in  the  more  usual  mag- 
netic calculations,  is  used  to  measure  the  magnetomotive  force  produced 
per  unit  length  of  a  coil  or  solenoid  carrying  an  electric  current ;  or  the 
magnetomotive  force  required  per  unit  length  of  any  part  of  a  magnetic 
circuit  to  produce  the  desired  flux  density  in  that  part.  The  usual  calcu- 
lations of  magnetic  circuits  then  often  become  simpler  than  they  would  be 
if  the  whole  magnetomotive  force  itself  is  used.  In  electric  circuits  it  has 
its  analogy  in  the  electromotive  force  produced  per  centimeter  length  of 
active  wire  in  an  armature  of  a  dynamo  or  in  a  transformer,  or  the  difference 
of  potential  required  per  centimeter  length  of  a  conductor  in  order  to  pro- 
duce the  desired  current  density  in  that  conductor. 

There  are  no  specifically  named  units  of  magnetizing  force.  The  one 
most  frequently  used  when  the  metric  system  is  employed,  is  an  ampere- 
turn  per  centimeter  length.  When  inches  are  used  the  unit  is  an  am- 
pere-turn per  inch.  The  absolute  or  C.G.S.  unit  is  one  gilbert  per 
centimeter.  When  the  magnetic  circuit  consists  of  air,  the  magnetizing 

t  Or  10H-4;r.     Aprx.  %0.     Log  1-900  7901- 
i  Or  l-*-4ff.     Aprx.  8-s-lOO.     Log  2-900  7901- 


MAGNETIZING    FORCE.  135 

force  can  also  be  expressed  and  measured  in  terms  of  units  of  flux  density, 
namely  gausses,  for,  although  they  mean  something  different,  they  are 
numerically  the  same  as  gilberts  per  centimeter  for  air,  as  is  shown  below. 

If  a  current  flows  through  a  uniform  coil  of  wire  which  is  very  long  as 
compared  with  its  diameter,  the  flux  density  produced  in  its  interior  will 
be  practically  uniform  except  near  its  ends;  it  will  be  nearly  uniform 
throughout  if  the  two  ends  are  brought  together  to  form  a  ring  coil.  More^ 
over,  this  flux  density  is  independent  of  the  diameter  of  the  coil  or  the  shape 
of  its  cross-section,  which  affect  only  the  reluctance  and  the  total  flux. 

The  magnetizing  force  produced  by  such  a  coil,  whether  it  contains  iron 
or  not,  is  numerically  equal  in  gilberts  per  centimeter  to  4irnc,  in  which  n 
is  the  number  of  turns  or  windings  per  ceritimete  length  of  coil,  and  c  is  the 
current  in  absolute  units;  nc  is  therefore  the  number  of  current-turns  per 
centimeter,  corresponding  to  (but  not  numerically  equal  to)  the  ampere- 
turns  per  centimeter.  Imagine  a  series  of  planes  perpendicular  to  the  axis 
and  one  centimeter  apart;  then  the  magnetizing  force  given  by  this  formula 
will  be  the  magnetomotive  force  in  gilberts  produced  between  each  plane 
and  the  next.  As  an  analogy,  suppose  the  interior  were  replaced  by  a  con- 
ductor carrying  an  electric  current,  and  the  coil  itself  were  replaced  by  a 
device  which  induces  an  electromotive  force  in  that  conductor,  then  the 
volts  induced  per  centimeter  length  of  this  device  will  evidently  be  the 
volts  of  electromotive  force  which  exist  between  each  of  these  parallel  planes 
and  the  next. 

It  is  also  true  for  iron  as  well  as  for  air  that  the  magnetizing  force  equals 
the  flux  density  divided  by  the  permeability,  but  as  the  permeability  of 
air  is  unity  by  definition,  it  follows  that  for  air  (or  more  exactly  for  a 
vacuum)  the  magnetizing  force  of  such  a  coil  when  expressed  in  gilberts 
per  centimeter,  is  numerically  equal  to  the  flux  density  in  its  interior  in 
gausses  The  above  general  formula  therefore  also  gives,  as  a  special  case, 
the  flux  density  in  air  in  gausses.  This  has  given  rise  to  much  confusion,  as 
it  makes  it  appear  at  first  sight  as  tlwugh  a  magnetomotive  force  was  of 
the  same  nature  as  a  flux  density,  which  with  the  electric  units  would  be 
like  saying  that  an  electromotive  force  was  of  the  same  nature  as  a  current 
density.  The  explanation  is  that  the  reluctance  of  a  centimeter  cube  of 
air  is  unity,  hence  the  flux  density  in  gausses  through  each  centimeter  cube 
of  air  will  be  numerically  the  same  as  the  magnetomotive  force  in  gilberts 
acting  between  its  two  opposite  faces;  or  in  the  above  illustration  with 
the  imaginary  parallel  planes  one  centimeter  apart,  the  flux  density  in 
gausses  in  each  space  between  two  such  planes  will  for  air  be  numerically 
equal  to  the  magnetomotive  force  (in  gilberts)  between  each  plane  and  the 
next.  Analogously,  if  a  wire  happens  to  have  a  resistance  of  one  ohm  per 
foot,  the  number  of  volts  acting  at  the  ends  of  each  foot  will  be  numerically 
the  same  as  the  number  of  amperes  flowing.  The  identity  is  in  the  num- 
bers and  not  necessarily  in  the  nature  of  the  units;  moreover,  the  numerical 
identity  exists  only  between  the  units  gilberts  per  centimeter  and  gauss, 
and  not  between  the  other  units  like  ampere-turns. 

The  formula  4?mc  therefore  always  gives  the  magnetizing  force  produced 
in  gilberts  per  centimeter  length  of  coil  .whether  there  is  iron  in  the  coil  or 
not.  It  also  gives  the  flux  density  in  gausses  in  the  interior  of  the  coil, 
but  for  air  only,  c  is  the  current  in  absolute  units  and  must  be  replaced 
in  the  formula  by  C-^10,  if  C  is  to  be  in  amperes.  When  the  result  given 
by  this  formula  is  multiplied  by  the  entire  length  of  the  coil  in  centi- 
meters, it  gives  the  total  magnetomotive  force  of  the  whole  coil,  in  gilberts. 

The  same  magnetizing  force  will  produce  entirely  different  flux  densities 
in  materials  of  different  permeabilities,  just  as  the  same  voltage  will  pro- 
duce entirely  different  currents  in  materials  of  different  conductivities. 
The  magnetizing  force  in  gilberts  per  centimeter  multiplied  by  the  permea- 
bility of  any  material  gives  the  flux  density  in  gausses,  or  the  induction  in 
gausses,  produced  in  that  material  by  that  magnetizing  force.  Or  in  dif- 
ferent terms,  if  a  coil  produces  in  its  interior  a  certain  flux  density  in  gausses 
in  air,  then  that  flux  density  must  be  multiplied  by  the  permeability  of  the 
material  to  get  the  flux  density  or  induction  in  that  material  in  gausses, 
when  the  air  circuit  is  completely  replaced  by  that  material,  as  in  trans- 
formers; for  the  case  in  which  the  circuit  is  partly  air  and  partly  iron,  see 
the  next  paragraphs.  The  more  usual  calculations,  such  as  those  for  dy- 
namos and  transformers,  start  with  the  desired  induction,  and  in  order  to 
avoid  the  calculation  of  the  reluctance,  the  troublesome  factor  0.4*,  and 


136 


MAGNETIZING    FORCE. 


the  various  other  reduction  factors  when  inch  units  are  used,  such  calcula- 
tions are  generally  made  as  described  in  the  following  paragraphs. 

Magnetic  calculations,  like  those  for  dynamos  and  transformers.  The 
values  of  the  permeabilities  of  the  particular  iron  or  steel  which  is  to  be 
used,  are  usually  given  in  the  form  of  a  curve  called  a  permeability  curve 
or  a  magnetization  curve,  whose  horizontal  distances  are  the  magnetizing 
forces  in  ampere-turns  per  centimeter  (//),  and  whose  vertical  ones  are  the 
corresponding  inductions  (B)  or  flux  densities,  in  gausses  or  lines  of  force 
per  square  centimeter  in  that  quality  of  iron  or  steel.  It  must  be  decided 


multiply  this  by  the  axial  or  center-line  length  in  centimeters,  of  this  par- 
ticular part  of  the  iron  under  consideration,  be  it  the  cores  or  the  yoke- 
pieces,  or  the  armature,  and  the  result  will  be  the  total  number  of  ampere- 
turns  or  the  magnetomotive  force,  which  is  required  to  magnetize  that  part 
to  that  particular  induction.  Notice  that  the  length  here  used  is  that  of 
the  path  of  the  flux  through  that  part  of  the  iron,  and  not  necessarily  the 
length  of  the  coil.  Having  done  this  for  each  of  the  iron  parts  making  up 
the  whole  circuit,  add  them  all  together  and  the  result  will  be  the  total 
ampere-turns  required  for  the  total  iron  part  of  the  circuit.  The  ampere- 
turns  required  for  producing  the  flux  in  the  air-gap  are  calculated  from  the 
desired  flux  density  in  the  gap,  as  follows:  ampere-turns  =  flux  density  in 
gausses X length  of  air-path  of  flux  (that  is,  twice  the  length  of  the  gap)  in 
centimeters  X 0.795  775.  These  latter  ampere-turns  (which  generally  are 
by  far  the  larger  part  of  the  whole)  are  then  added  to  those  required  for  the 
iron  part,  thus  giving  the  total  required  for  the  complete  magnetic  circuit. 
The  coils  for  producing  these  ampere-turns  may  have  any  length  and  may 
be  wound  around  any  convenient  part  of  the  circuit,  for  when  the  mag- 
netic circuit  is  chiefly  of  iron,  the  magnetizing  force  of  the  coils  will  act  in 
that  circuit  very  nearly  the  same  way  no  matter  how  they  are  distributed 
over  the  iron. 

When  the  dimensions  are  all  in  inches  and  the  flux  densities  in  maxwells 
per  square  inch  (inch-gausses),  then  the  magnetization  curve  should  be 

E lotted  for  those  units  and  the  calculations  are  then  precisely  the  same  except 
:>r  the  air-gap,  for  which  the  formula  then  becomes:  ampere-turns  =  flux 
density  in  maxwells  per  sq.  inch  X  length  of  air-path  of  flux  in  inches  X 
0.313  296. 

When  the  linear  dimensions  are  in  inches,  and  the  flux  densities  in  gausses, 
the  curve  should  be  plotted  accordingly  and  the  calculations  are  again  the 
same  except  for  the  air-gap,  for  which  the  formula  then  becomes:  ampere- 
turns  =  gausses X length  of  air-path  of  flux  in  inches  X2. 021  27. 

It  will  be  noticed  that  in  this  method  of  calculation  the  reluctance  need 
not  be  known.  The  total  required  cross-section  of  the  iron  is  determined 
by  dividing  the  given  total  flux  by  the  given  flux  density.  The  reverse 
calculation  to  the  above  is  much  more  difficult;  in  that  case  the  ampere- 
turns  of  such  a  composite  magnetic  circuit  together  with  all  the  dimen- 
sions are  given,  and  the  flux  produced  by  them  is  to  be  determined.  In 
such  a  case  perform  the  calculation  backwards,  by  making  trial  calcula- 
tions as  just  described,  using  different  assumed  total  fluxes,  until  one  is 
found  which  will  require  the  given  number  of  ampere-turns.  For  a  simple 
magnetic  circuit  like  that  in  a  transformer,  in  which  there  is  no  air-gap, 
either  calculation  amounts  to  little  more  than  reading  off  the  results  from 
the  magnetization  curve. 

Logarithm 
1  gilbert  per  inch: 

=  0.795  775  ampere-turn  per  inch.     Aprx.  subt.  % 1-900  7901 

=  0.393  700  gilbert  per  centimeter.     Aprx.  */io 1-595  1654 

=  0.313  296  ampere-turn  per  centimeter.     Aprx.  %e 1-4959555 

1  ampere-tuvii  per  inch: 

=    1.256  64  gilberts  per  inch.     Aprx.  add  % 0-099  2099 

=  0.494  738  gilbert  per  centimeter.     Aprx.  ^ 1-694  3753 

=  0.393  700  ampere-turn  per  centimeter.     Aprx- ^lo.  •  •  ....  1-595  1654 

1  gilbert  per  cm  =   2.540  01   gilberts  per  inch.     Aprx.  *% 0-404  8346 

=   2.02127   ampere-turns  per  inch.    Aprx.  2.  0-3056247 

1.  CGS  unit  (elmg) 0-000  0000 

=  0.795  775  amp.-turu  per  cm.    Ap.  subt.  %  .  L900  7901 


MAGNETIZING    FORCE. — MAGNETIC   FLUX.        137 

Lrgarithm 

1  CGS  unit  (elmg)  =  1.  gilbert  per  centimeter .  0-000  0000 

1  ampere-turn  per  cm: 

=  3.191  86  gilberts  per  inch.     Aprx.  3% 0-504  0445 

=  2.540  01  ampere-turns  per  inch.     Aprx.  1()4 0-404  8346 

=  1.256  64  gilberts  per  cm.     Aprx.  add  % 0-099  2099 

1  CGS  unit  of  current-turn  per  centimeter: 

=  31.918  6  gilberts  per  inch.     Aprx.  32 1.504  0445 

=  25.400  1  ampere-turns  per  inch.     Aprx.  25 1-404  8346 

=  12.566  4  gilberts  per  centimeter.     Aprx.  l/s  X  100 1-099  2099 

10.  ampere,yturns  per  centimeter 1-000  0000 

For  air  (or  vacuum)  only : 
1  gauss  =  1  gilbert  per  centimeter. 

The  relations  to  other  measures  are  as  follows : 
Ampere-turns  per  centimeter: 

=  gausses  X  0.795  77 5t  •*•  permeability. 

=  maxwells  X  0.7  95  77  5f-Ksq.  cm  section  X  permeability). 

=  gausses  X  0.7 95  77  5f-     For  air. 

=  maxwells  X  0.795  77  5f  •*-  sq.  cm  section.     For  air. 
Ampere-turns  per  inch: 

=  gausses  X  2.021  27  -*•  permeability. 

=  maxwells  X  0.313  297 -r-(sq.  inches  section  X  permeability). 

=  inch  gausses X 0.313  296  ^permeability. 

=  gausses  X  2. 021  27.     For  air. 

=  maxwells  X  0.313  297  -s-  sq.  inches  section.     For  air. 

=  inch  gausses  X  0.3 13  296.     For  air. 
Gilberts  per  centimeter  =  gausses  -f- permeability. 

=  max  wells -r-(sq.  cm  section  X  permeability). 
=  gausses.     For  air. 
=  maxwells  -r-  sq.  centimeters.     For  air. 
Gilberts  per  inch: 

=  gausses  X  2.540  01  -4-  permeability. 

=  maxwells  X  0.393  700  -5-  (sq.  inches  section  X  permeability). 

=  inch  gausses  X  0.393  700  -*•  permeability. 

=  gausses  X  2.540  01 .     For  air. 

=  maxwells  X  0.393  700  -4-  sq.  inches  section.     For  air. 

=  inch  gausses  X  0.393  700.     For  air. 
CGS  unit  of  current-turns  per  centimeter: 

=  gausses  X  0.079  577  5  Impermeability. 

=  gausses  X  0.079  577  54     For  air. 


MAGNETIC  FLUX  [*,  <£];  LINES  OF  FORCE;  FLUX 
OF  FORCE;  AMOUNT  OF  MAGNETIC  FIELD; 
POLE  STRENGTH  [m].  (Magnetomotive  force  -=-  reluct- 
ance ;  magnetic  induction  (or  flux  density)  X  surface  ;  elec- 
tromotive force X time;  length X \/f ore e.) 

These  units  are  used  to  measure  the  total  quantity  or  amount  or  number 
•f  magnetic  lines  of  force  or  the  amount  of  flow  or  flux  of  magnetism,  just 
as  amperes  are  used  to  measure  the  quantity  or  amount  of  electric  current, 
or  as  cubic  feet  per  second  measure  the  amount  of  a  flow  of  water.  This 
magnetic  flux  or  flow  is  in  some  respects  analogous  to  a  current  of  electricity, 
as  it  must  always  form  a  closed  circuit  upon  itself,  and  be  the  same  in 
amount  in  every  cross-section  of  the  circuit;  it  follows  a  law  analogous  to 
Ohm's  law,  as  the  flux  is  equal  to  the  magnetomotive  force  divided  by  the 

t  Or  10-^4*.     Aprx.  8/10.     Log  1. 900  7901- 
t  Or  l-fr-4*.     Aprx.  8/100.     Log  2-900  7901- 


138 


MAGNETIC    FLUX. 


reluctance.  It  differs,  however,  in  that  no  work  is  being  done  continuously 
in  the  circuit  of  a  magnetic  flux,  and  it  can  therefore  continue  to  exist  in- 
definitely as  in  a  permanent  magnet,  without  consuming  or  producing 
energy.  Energy  is  stored  in  the  flux  when  it  is  produced,  and  it  is  given  out 
again  whenever  the  flux  ceases  to  exist,  but  no  energy  is  necessarily  required 
to  maintain  it ;  it  is  therefore  in  this  respect  more  like  a  mechanical  pressure 
or  stress,  as  that  of  compressed  air.  The  kinetic  energy  (which  see  above) 
required  to  start  an  electric  current  is  stored  in  the  system  as  magnetic 
energy,  and  given  out  again  as  electrical  energy  when  the  current  stops. 
From  the  energy  standpoint ,  magnetic  flux  is  more  analogous  to  coulombs 
of  electricity.  A  magnetic  circuit  also  differs  from  an  electric  circuit  in 
that  it  can  never  be  opened,  as  there  is  no  such  a  thing  as  a  magnetic  insu- 
lator; magnetic  flux  can  cease  only  by  contracting  to  a  point  somewhat 
like  an  extremely  small  rubber  band  which  is  allowed  to  contract  after  hav- 
ing been  stretched. 

The  space  surrounding  a  magnet  or  an  electric  current  or  that  between 
two  magnetic  poles,  is  called  a  magnetic  field  or  magnetic  iield  of  force, 
as  it  contains  magnetic  flux  or  magnetic  lines  of  force;  units  of  flux  measure 
the  total  amount  of  this  field  (but  riot  its  intensity  or  density)  or  the  total 
number  of  such  lines  of  force ;  this  flux  also  continues  through  the  magnet 
itself. 

Flux  is  also  equal  to  the  flux  density  (sometimes  called  induction),  in 
lines  of  force  per  square  centimeter  (or  per  square  inch),  multiplied  by  the 
number  of  square  centimeters  (or  square  inches)  cross-section  of  its  path, 
and  is  then  often  called  the  total  flux  to  distinguish  it  from  the  flux  density. 
An  electric  current  is  always  encircled  by  such  flux,  and  the  two  circuits, 
namely  the  electric  and  the  magnetic,  are  always  linked  together  like  the 
two  links  of  a  chain.  When  the  magnetic  flux  enclosed  in  a  coil  or  loop  of 
wire  is  increased  or  diminished,  an  electromotive  force  is  produced  in  that 
wire;  this  is  the  fundamental  principle  of  a  dynamo;  or  stated  in  different 
terms,  when  a  wire  cuts  through  magnetic  flux,  an  electromotive  force  is 
produced  in  the  wire;  or  the  linking  and  unlinking  of  circuits  of  flux  and 
electric  circuits  produces  an  electromotive  force;  it  is  this  that  produces 
self-  and  mutual  induction. 

The  term  flux-turns  or  mean  flux-turns  is  sometimes  used  for  denot- 
ing the  product  of  the  number  of  turns  and  the  mean  flux  (in  maxwells)  in 
one  turn;  the  magnetic  leakage  is  thereby  eliminated.  The  mean  flux- 
turns  in  maxwell-turns  are  equal  to  the  self-inductance  in  henrys  multiplied 
by  108  times  the  final  current  in  amperes. 

The  unit  universally  used  is  the  absolute  or  electromagnetic  C.G.S. 
unit  or  single  line  offeree,  and  is  defined  as  that  amount  of  flux  which 
acting  on  a  unit  magnetic  pole  will  propel  it  with  a  force  of  one  dyne.  It 
can  also  be  defined  as  the  amount  of  flux  passing  through  one  square  centi- 
meter cross-section  of  a  field  having  a  flux  density  of  one  C.G.S.  unit.  A 
unit  magnetic  pole  (imaginary)  is  one  which  will  exert  a  force  of  one 
dyne  on  another  unit  pole  one  centimeter  distant.  From  each  such  pole 
there  radiates  a  flux  equal  to  4;r  (or  12.566  4)  of  these  units  or  lines  of  force. 

A  single  or  unit  line  of  force  in  a  magnetic  field  may  be  said  to  stand  for 
or  represent  a  tube  of  such  a  cross-section  that  it  always  embraces  a  unit 
of  flux;  a  definite  amount  of  flux  may  have  widely  different  lengths  of  cir- 
cuit or  cross-sections  without  changing  its  amount. 

This  C.G.S.  unit  is  called  a  maxwell,  according  to  the  International 
Congress  of  1900.  A  maxwell  is  therefore  the  same  thing  as  a  single  or 
unit  line  of  force  as  above  defined. 

Logarithm 

1  COS  unit  (elmg)  =  1  maxwell. 0  000  0000 

1  maxwell: 

1  CGS  unit  (elmg) 0  000  0000 

1  line  of  force. 0  000  OGOO 

1  gauss-centimeter2 0  000  0000 

=     0.155  000  gauss-inch2.    Aprx.  3ia 1-190  3308 

=  0.079  577  5  (or  34 TT)  of  the  flux  from  a  unit  pole.  Aprx.8/ioo  29007901 

1  weber  (obsolete)  =  1  maxwell 0-000  0000 

1  unit  pole  (flux  from)  =  12.566  37  (or  4 ?r)  maxwells 1.099  2099 

1  Kapp  line  (obsolete)  =  6  000.  maxwells 3  778  1513 


MAGNETIC   FLUX.  139 

The  relations  to  other  measures  are  as  follows: 
Maxwells: 

=  gausses  X  sq.  centimeters. 

=  in  -h-gaussesXsq.  inches. 

=  gilberts -r- oersteds. 

=  gilberts  X  permeance  (in  CGS  units)., 

=  gilberts  X  permeability  X  sq.  centimeters  section  -&•  centimeters  length. 

=  gilberts X  permeability  X  sq.  inches  section  X  2.540  01  -s- inches  length. 

=  gilberts  Xsq.  centimeters  section -r- centimeters  length.     For  air. 

=  gilberts Xsq.  inches  section X  2. 540  01  -f- inches  length.     For  air. 

=  gilberts  per  centimeter  X  permeability  X  sq.  centimeters  section. 

=  gilberts  per  inch  X  permeability  X  sq.  inches  section  X  2. 540  Oi. 

=  gilberts  per  centimeter  X  sq.  centimeters  section.     For  air. 

=  gilberts  per  inch  X  sq.  inch  section  X  2.540  01.     For  air. 

=  ampere-turns  X  1 .256  64f  -*-  oersteds. 

=  ampere-turnsX  1.256  64f  X  permeance  (in  CGS  units). 

=  ampere-turns  X  permeability  X  sq.  centimeters  section  X  1.25.664f-*- 
centimeters  length. 

=  ampere-turns  X  permeability  X  sq.  inches  section  X  3.191  86  -f- inches 
length. 

—  ampere-turns  X  sq.    centimeters    section  X  1 .256  64  f  •*•  centimeters 
length.     For  air. 

=  ampere-turns  Xsq.  inches  section  X  3. 191  86  ^-inches  length.     For  air. 

=  ampere-turns    per    centimeter  X  permeability  X  sq.    centimeters    sec- 
tion X  1.256  64.  f 

=  ampere-turns  per  inch  X  permeability  Xsq.  inches  section  X  3. 191  86. 

«=  ampere-turns  per  cm  X  sq.  centimeters  section  X  1.256  64.  t  v  For  air. 

=  ampere-turns  per  inchXsq.  inches  section  X  3. 191  86.     For  air. 

=  CGS  unit  current-turns  X  12.566  4J -5- oersteds. 

(For  further  relations  with  CGS  unit  current-turns,  multiply  those  in 
terms  of  ampere-turns  by  10;  that  is,  substitute  for  "ampere-turns"  in 
any  of  the  above,  the  quantity  "CGS  unit  current-turns X  10.") 

Maxwells  =  volts  X  seconds  XIO8-^  number  of  turns. 

=  volts  X  seconds  X  108.     For  a  single  conductor. 

When  the  flux  is  due  only  to  the  current,  as  in  self-induction,  and  when 
there  is  no  magnetic  leakage: 
'Maxwells  =  joules  of  stored  energy  X  2  X  10s  -j-  ampere-turns. 

=  joules  of  stored  energy  X2X  108 -s- amperes.     For  a  single  wire. 
=  ergs  of  stored  energy  X  20 -f- ampere-turns. 
=  henrys  X  ampere-turns  X  108  -*-  number  of  turns2. 
"  =  henrys  X  final  amperes  X  108 -5- number  of  turns. 

When  there  is  magnetic  leakage,  substitute  for  "maxwells"  in  the  above, 
"mean  maxwells."  The  mean  maxwells  are  the  mean  flux-turns  divided 
by  the  total  number  of  turns. 

When  the  flux  is  from  an  external  source,  and  independent  of  the  current, 
as  in  mutual  induction,  and  when  there  is  no  magnetic  leakage: 
Max  wells  =  joules  of  stored  energy  X  108-r-  ampere-turns. 

=  joules  of  stored  energy  X  10s  -f-  amperes.     For  a  single  conductor. 
=  ergs  of  stored  energy  X  10-4- ampere-turns. 

When  there  is. magnetic  leakage,  make  the  same  substitution  as  above 
described. 
Mean  maxwells  (through  the  secondary): 

=  henrys   (of  mutual  induction)  X  final  amperes   (of  primary)  X  108  •+• 
number  of  turns  (of  secondary). 


t  Or  47T/10.     Aprx.  add  l/i.     Log  0-099  209fJ. 
t  Or  4n.     Aprx.  1/%X  100.     Log  1.Q99  2099- 


140 


MAGNETIC    FLUX   DENSITY. 


MAGNETIC  FLUX  DENSITY  [X,  H]j  .  MAGNETIC 
INDUCTION  [®;  B] ;  LINES  OF  FORCE  PER  UNIT 
CROSS-SECTION;  EARTH'S  FIELD.  (Flux^-sur- 
face ;  magnetizing  force  X  permeability.) 

This  quantity,  which  is  one  of  the  most  important  in  magnetic  calcula- 
tions, measures  the  extent  to  which  a  body  is  magnetized  as  expressed  by 
the  amount  of  flux  which  exists  per  square  centimeter  (or  square  inch)  of 
cross-section  of  the  circuit;  it  gives  the  density  of  the  flux  in  C.G.S.  units 
(maxwells  or  lines  of  force)  per  square  centimeter  or  square  inch  cross-sec- 
tion. It  corresponds  to  current  density  in  electrical  calculations.  As  it 
specifies  or  determines  the  strength  of  a  magnetic  field ,  it  is  often  called 
the  field  strength  or  field  intensity  ;t  the  earth's  magnetic  field,  for  instance, 
is  expressed  in  these  units.  When  it  refers  to  air  it  is  generally  represented 
by  3C  or  simply  H.  When  it  refers  to  the  flux  density  "induced"  in  a  mag- 
netic material,  such  as  iron,  by  an  outside  source,  such  as  a  current  in  a  coil 
of  wire,  it  is  often  called  the  induction,  generally  expressed  by  (B  or  sim- 
ply by  B,  which  is  one  of  the  most  important  quantities  in  magnetic  cal- 
culations. In  the  calculation  and  design  of  dynamos  and  transformers 
this  induction  or  flux  density  in  the  iron  is  of  prime  importance.  From  it 
as  a  starting-point,  the  size  of  the  core  and  the  required  ampere-turns  are 
calculated,  as  was  explained  briefly  under  the  units  of  magnetizing  force. 
The  saturation-point  of  magnetic  material  like  iron  is  expressed  in  terms 
of  these  units  of  flux  density.  The  total  flux  in  maxwells  is  equal  to  the 
flux  density  in  gausses  multiplied  by  the  total  cross-section  in  square  centi- 
meters. As  the  permeability  or  reluctivity  of  air  is  unity,  it  follows  that 
the  magnetizing  force  in  gilberts  per  centimeter  of  a  long  coil  is  numerically 
the  same  as  the  flux  density  in  gausses  produced  by  it  in  the  interior  of  the 
coil,  when  there  is  no  magnetic  material  in  it.  For  this  reason  the  magne- 
tizing force  is  often  confused  with  flux  density  or  its  equivalent  the  intensity 
of  field,  as  it  is  then  often  called.  This  applies  only  to  the  absolute  units; 
when  ampere-turns  or  when  inch  units  are  used,  a  numerical  factor  must  be 
introduced. 

The  unit  universally  used  when  the  dimensions  are  in  the  metric  system 
is  the  electromagnetic  C.G.S.  unit,  which,  according  to  the  International 
Congress  of  1900,  is  called  a  gauss.  It  is  defined  as  that  field  intensity 
which  is  produced  at  the  center  of  a  circle  of  one  centimeter  radius  by  1 
C.G.S.  unit  of  current  (or  10  amperes)  flowing  through  an  arc  of  this 
circle  one  centimeter  long.  This  is  one  of  the  relations  which  connect 
the  electric  with  the  magnetic  units.  It  can  also  be  defined  as  the  inten- 
sity of  the  field  at  one  centimeter  distance  from  a  unit  pole,  that  is,  at  the 
surface  of  a  sphere  of  one  centimeter  radius,  having  an  imaginary  C.G.S. 
.unit  pole  at  its  center;  from  such  a  pole  4;r  unit  lines  of  force  (maxwells) 
emanate,  and  as  the  area  of  the  sphere  is  4?r  square  centimeters,  it  follows 
that  the  flux  density  will  be  one  line  of  force  or  maxwell  per  square  centi- 
meter of  the  spherical  surface.  It  may  also  be  defined  as  that  field  inten- 
sity which  will  exert  a  pull  of  one  dyne  on  an  (imaginary)  isolated  unit 
magnetic  pple  placed  in  it.  All  three  of  these  definitions  refer  to  the  same 
unit.  The  formulas  giving  the  field  intensity  in  coils,  like  those  for  mag- 
nets or  for  galvanometers,  all  give  the  result  in  terms  of  this  unit,  provided 
the  current  is  stated  in  terms  of  the  absolute  unit  of  current  which  is  equal 
to  10  amperes;  great  care  must  be  taken  in  such  formulas  to  use  the  proper 
unit  of  current;  it  should  always  be  stated  whether  the  formula  has  been 
reduced  to  amperes  or  not. 

The  practical  unit  is  therefore  the  same  as  the  C.G.S.  unit,  namely 
the  gauss,  which  means  one  maxwell  (or  line  of  force)  per  square  centi- 
meter; and  a  flux  density  or  induction  stated  in  a  number  of  gausses 
means  that  number  of  maxwells  (or  lines  of  force)  per  square  centimeter. 

fit  should  be  distinguished,  however,  from  the  term  intensity  of  mag- 
netization (see  below),  which  is  a  term  sometimes  used  in  physics  and  has  a 
different  meaning. 


MAGNETIC    FLUX   DENSITY.  141 

When  the  dimensions  are  in  inches  the  flux  density  and  induction  are 
often  for  convenience  stated  in  lines  of  force  (or  maxwells)  per  square  inch; 
this  in .h  unit  has  no  generally  accepted  name;  the  name  inch  gauss  is 
here  proposed  and  is  used  in  these  tables. 

Logarithm 

I  maxwell  per  sq.  inch  =  1.  inch  gauss 0-000  0000 

=  0.155000  gauss.     Aprx.  2,4s 1-1903308 

1  inch  gauss  =  1.  maxwell  per  sq.  inch 0  000  0000 

=  0.155  000  gauss.     Aprx.  2A3 ]  .190  3308 

1  CGS  unit  (elmg)  =  1 .  ga.uss 0  000  0000 

1  gauss  =  6. 451  63  maxwells  per  sq.  inch.    Aprx.  x%  or  6^.  .  .  .    0-809  6692 

=  6.451  63  inch  gausses.  .  Aprx.  1:%  or  6^ 0  809  6692 

=  1.  CGS  unit  (elmg)  of  flux  density 0  000  0000 

1.  CGS  unit  (elmg)  of  flux  per  sq.  centimeter.  .  .    0-000  0000 

=  1.  maxwell  per  sq.  centimeter 0  000  0000 

1.  magnetic  "line  of  force"  per  sq.  centimeter..  .  0  000  0000 

1  maxwell  per  sq.  centimeter  =  1.  gauss 0  000  0000 

1  kilogauss  =1  000.  gausses 3-000  0000 

The  relations  to  other  measures  are  as  follows: 
Gausses  =  max  wells  -r-sq.  centimeters. 
=  inch  gausses  X  0.1 55  000. 
=  gilberts  per  centimeter  X  permeability. 
=  gilberts  per  inch  X  permeability  X  0.393  700. 
=  gilberts  per  centimeter.     For  air. 
-gilberts  per  inch  X  0.393  700.     For  air. 
=  gilberts  X  permeability  -r-  centimeters. 
=  gilberts  X  permeability  X  0.393  700  -5- inches. 
=  gilberts -=- centimeters.     For  air. 
=  gilberts  X  0.393  700  H- inches.     For  air. 
=  gilberts  -r-  (oersteds  X  sq.  centimeters). 
=  ampere-turns  per  centimeter  X  permeability  X  1.256  64. f 
=  ampere-turns  per  inch  X  permeability  X  0.494  738. 
=  ampere-turns  per  centimeter  X  1.256  64. f     For  air. 
=  ampere-turns  per  inchXO.494  738.     For  air. 
=  ampere-turns  X  permeability  X  1  -256  64 1  •*•  centimeters. 
=  ampere-turns  X  permeability  X  0.494  738  -=- inches. 
=  ampere-turns  X  1 .256  64|  -*- centimeters.     For  air. 
=  ampe re-t urns  X  0.494  738  -i- inches.      For  air. 
=  ampere-turns  X  1 .256  64t  •*-  (oersteds  X  sq.  centimeters). 
=  CGS  unit  current-turns  per  centimeter  X  permeability  X  12.566  4.t 
(For  further  relations  with  CGS  unit  current-turns,  multiply  those  in 
terms  of  ampere-turns  by  10;    th -t  is,  substitute  for  "ampere-turns"  in 
any  of  the  above,  the  quantity  "CGS  unit  current-turns X  10.") 
Gausses  =  volts  X  seconds  X  10s  -r-  (number  of  turns  X  sq.  centimeters). 

=  CGS  units  of  intensity  of  magnetization  X  12.566  44 
Inch  gausses : 

=  max  wells -i-sq.  inches. 

=  gausses  X  6. 451  63. 

=  gilberts  per  centimeter  X  permeability  X  6.451  63. 

=  gilberts  per  inch  X  permeability  X  2. 540  01. 

—  gilberts  per  centimeter  X  6.451  63.     For  air. 

=  gilberts  per  inch  X  2.540  01 .     For  air. 

=  gilberts  X  permeability  X  2.540  01  -s-  inches. 

=  gilberts  X  2.540  01  ^-inches.     For  air. 

=  gilberts -*-  (oersteds  X  sq.  inches). 

=  ampere-turns  per  centimeter  X  permeability  X  8. 107  35 

=  ampere-turns  per  inch  X  permeability  X  3.191  86. 

=  ampere-turns  per  centimeter  X  8. 107  35.     For  air. 

=  ampere-turns  per  inch  X  3.191  86.     For  air. 

=  amoere-turns  X  permeability  X  3.191  86  -Cinches. 

=  ampere-turns X 3. 191  86-r-inches.     For  air. 

=  ampere-turns  X  1.256  64f-=-(  oersteds  X  sq.  inches). 

=  CGS  unit  current-turns  per  centimeter  X  permeability  X  8 1.07 3  5. 


t  Or  4;r/10.     Aprx.  add  M-     Log  0  099  2099, 
JOr4r.     Aprx.  i^XlOO.     Log  1  099  2099- 


142  MAGNETIC   MOMENT.  —  INTENSITY. 

(For  further  relations  with  CGS  unit  current-turns,  multiply  those  in 
terms  of  ampere-turns  by  10;  that  is,  substitute  for  "ampere-turns"  in 
any  of  the  above  the  quantity  "CGS  unit  current-turns  X  10.") 
Inch  gausses  =  volts  X  seconds  X  10s  -s-  (number  of  turns  X  sq.  inches). 
Gausses  in  iron  =  gausses  in  air  X  permeability. 

'  '         in  air    =  gausses  in  iron  -5-  permeability. 
Inch  gausses  in  iron  =  inch  gausses  in  air  X  permeability. 
in  air    =  inch  gausses  in  iron  -*-  permeability. 

MAGNETIC   MOMENT   [fflfe].     (Pole  strength  X  length.) 

This  quantity,  used  chiefly  in  magnetometry,  is  the  product  of  the  pole 
strength  of  a  magnet  multiplied  by  its  theoretical  length,  that  is,  by  the 
distance  between  the  two  centers  at  which  the  poles  may  be  considered  to 
be  condensed.  As  a  pole  (see  under  flux)  is  not  measured  in  units  like 
force,  such  a  moment  is  not  directly  comparable  with  a  mechanical  momer  t 
called  torque,  but  as  the  force  existing  between  two  unit  poles  one  centi- 


meter apart  is  one  dyne,  a  magnetic  moment  may  be  converted  into  a 
mechanical  moment.  As  a  single  pole  has  no  real  existence  such  a  cal- 
culation in  practice  always  involves  the  action  of  two  poles  on  two  others. 


. 

There  are  no  special  units.  The  C.G.S.  unit  is  a  unit  pole  multiplied 
by  a  centimeter,  and  would  therefore  be  called  a  pole-centimeter.  A  max- 
well-centimeter might  also  be  used,  as  a  unit  pole  has  4^  (  =  about  12^) 
maxwells  or  lines  of  force  issuing  from  it  ;  each  line  of  force  exerts  a  force 
of  one  dyne  on  a  unit  pole.  Two  unit  poles  one  centimeter  apart  attract 
or  repel  each  other  with  a  force  of  one  dyne;  the  force  between  any  two 
poles  is  proportional  to  the  product  of  the  two  pole  strengths  in  terms  of 
the  above  unit  poles,  and  inversely  proportional  to  the  square  of  the  dis- 
tance between  them  in  centimeters. 
1  unit-  pole-centimeter  unit  -=1.  unit  pole  XI.  centimeter. 

=  1.  CGS  unit  of  magnetic  moment. 
The  relation  to  other  measures  are  as  'follows: 
Magnetic  moments  (in  CGS  units): 
=  CGS  unit  poles  X  centimeters. 

=  intensity  of  magnetization  (in  CGS  units)  X  volume  in  cb.  cm. 
=  gausses  X  0.07  9  577  5  X  volume  in  cb.  cm. 

INTENSITY  OF  MAGNETIZATION  [3,  I];  MOMENT 
PER  UNIT  VOLUME;  POLE  STRENGTH  PER 
UNIT  CROSS-SECTION.  (Magnetic  moment  4-  volume  ; 
pole  strength  -f-  surface  .) 

This  quantity,  used  chiefly  in  physical  conceptions,  measures  the  polar- 
'.zed  state  of  the  interior  of  a  magnet.  If  a  magnet  were  cut  into  small 
pieces  (assuming  that  the  magnetic  state  was  not  altered  thereby)  each 
piece  would  be  a  separate  magnet  whose  magnetic  moment  bears  the  same 
proportion  to  its  volume  as  the  moment  of  the  original  magnet  bears  to 
its  volume,  hence  the  magnetic  state  remains  the  same  if  stated  in  the  mag- 
netic moment  per  cubic  centimeter,  which  quantity  is  called  the  intensity 
of  magnetization.  It  is  also  the  pole  strength  per  square  centimeter  cross- 
section.  As  pole  strength  is  convertible  into  flux  (maxwells),  it  follows 
that  the  intensity  of  magnetization  is  a  unit  of  the  same  nature  as  flux  den- 
sity, that  is,  maxwells  per  square  centimeter  or  gausses.  They  differ  only 
in  the  bases  on  which  they  are  defined. 

The  C.G.S.  unit  is  one  unit  moment  per  cubic  centimeter,  that  is,  one 
unit-pole-centimeter  per   cubic   centimeter,   or  one   unit   pole   per   square 
centimeter.     This  unit  is  numerically  equal  to  4n  gausses,  as  its  relation  to 
gausses  is  the  same  as  the  relation  of  a  unit  pole  is  to  a  unit  of  flux. 
1  CGS  unit  of  intensity  of  magnetization: 

1.  CGS  unit  of  magnetic  moment  per  cb.  centimeter. 

=  1.  unit-pole-centimeter  unit  of  magnetic  moment  per  cb.  cm. 

=  12.566  4  (or  4;r)  gausses. 
The  relations  to  other  measures  are  as  follows: 
CGS  units  of  intensity  of  magnetization: 

=  CGS  units  of  magnetic  moments  -^cb.  centimeters. 

=  gausses  X  0.07  9  577  5. 


MAGNETIC    ENERGY.  143 


MAGNETIC   WORK  or  ENERGY  [W],      (Magnetomotive 
force  X  flux  5  ampere-turns  X  flux.) 

This  quantity  is  seldom  used  in  calculations.  When  a  current  is  started 
in  a  wire  or  in  an  electro-magnet,  or  when  the  armature  of  a  steel  magnet 
is  pulled  off,  magnetic  energy  is  stored ;  it  is  given  out  again  in  some  other 
form,  often  in  the  form  of  a  spark,  when  the  current  is  stopped,  or  as 
mechanical  energy  when  a  permanent  magnet  attracts  its  armature  to 
itself.  In  a  transformer,  the  energy  of  the  primary  current  is  all  converted 
into  magnetic  energy  which  is  reconverted  into  electrical  energy  in  the 
secondary  circuit.  The  magnetic  energy  is  equal  to.  and  in  fact  is  the  same 
thing  as,  the  kinetic  energy  of  a  current  (which  see  above).  It  is  equal  to 
the  product  of  magnetomotive  force  and  flux.  It  appears  as  heat  in  the 
hysteresis  loss.  It  should  not  be  confused  with  the  power  used  contin- 
uously in  exciting  an  electromagnet,  as  that  power  is  all  converted  electric- 
ally into  heat;  it  is  only  when  the  current  is  first  started  that  any  electric 
energy  is  converted  into  magnetic  energy.  Magnetic  flux  itself  is  not 
energy  any  more  than  coulombs  of  electricity  or  mechanical  pressure; 
energy  is  required  to  produce  a  pressure,  but  not  necessarily  to  maintain 
it,  and  so  it  is  with  magnetic  flux  (which  see  above). 

There  are  no  specific  units  of  magnetic  energy;  it  is  usually  measured  in 
joules  or  ergs,  but  may  be  measured  in  terms  of  any  of  the  units  of  energy, 
which  see.  The  C.G.S.  unit  is  the  erg;  the  practical  unit  is  the  joule. 

The  relations  to  other  measures  are  as  follows: 
Joules  of  magnetic  energy  [J] 
=  henrys  X  final  amperes2  -*-  2. 
=  henrys  X  applied  volts2 -*-  (ohms2 X  2). 
=  time  constant  in  seconds  X  ohms  X  final  amperes2 -r- 2. 
=  time  constant  in  seconds  X  final  amperes  X  applied  volts  •*•  2. 
=  time  constant  in  seconds  X  applied  volts2 -5- ( ohms  X  2). 
When  the  flux  is  due  only  to  the  current,  as  in  self-induction,  and  wheix 
there  is  no  magnetic  leakage: 
Joules  of  magnetic  energy: 

=  max  wells  X  ampere-turns  -f-  (2  X  108). 

=  gausses  X  sq.  centimeters  X  ampere-turns  -*-  (2  X 108). 

=  inch-gausses  X  sq.  inches  X  ampere-turns -5-  (2  X  108). 

=  maxwells2X  oersteds X 0.397  887t-^-108. 

= maxwells  X  gilberts  X  0. 397  887 1  ^  108. 

= gilberts2  X  0.397  887 f-*-  (oersteds  X  10s). 

=  ampere-turns2  X  permeability  X  sq.  centimeter   section  X  0.628  318 

(or  2 TT /I ())•*-( centimeters  lengthXIO8). 
=  ampere-turns2  X  permeability  X  sq.   inches   section  X  1 .595  93  •*-  inches 

lengthXIO8. 

(For  further  relations  substitute  for  any  of  the  above  units  their  equiva- 
lents in  terms  of  the  desired  units,  as  given  in  the  other  tables.) 

When  the  flux  is  from  an  external  source  and  independent  of  the  current, 
as  in  mutual  induction,  and  when  there  is  no  magnetic  leakage,  the  mag- 
netic energy  is  twice  as  great  as  that  given  by  the  above  relations;  hence 
all  the  values  above  given  must  be  multiplied  by  2. 

When  there  is  magnetic  leakage,  use  the  "mean  maxwells"  instead  of 
the  "maxwells."  The  mean  maxwells  are  the  mean  flux  turns  divided  by 
the  total  number  of  turns. 

Ergs  of  magnetic  energy  =  henrys X final  amperes2X5X  10°. 
(For  further  relations  of  ergs  to  other  units  multiply  those  given  above 
for  joules  by  107.) 

t  Or  10-*-8;r.     Aprx.  <Ho.     Log  1.599  7601- 


144  MAGNETIC   POWER. 


MAGNETIC  POWER  [P],      (Magnetomotive  force X  flux -s- 
time  $  ampere-turns  X  flux  X  frequency.) 

This  quantity  is  seldom  used  in  calculations,  and  has  little  or  no  signifi- 
cance in  practice.  It  is  the  rate  at  which  magnetic  work  or  energy  is  per- 
formed (see  Magnetic  Energy  above);  it  is  therefore  equal  to  magnetic 
energy  divided  by  time.  It  is  met  with  in  practice  in  the  alternating  mag- 
netic fields  of  alternating  electric  currents,  for  in  these  the  magnetic  field 
is  continually  being  produced  at  a  rate  proportional  to  the  frequency, 
hence  it  is  proportional  to  the  product  of  the  magnetomotive  force,  the  flux, 
and  the  frequency.  The  power  of  the  primary  current  in  a  transformer  is 
transmitted  to  the  secondary  in  the  form  of  magnetic  power,  part  of  it 
being  lost  as  magnetic  power  in  the  form  of  hysteresis. 

With  alternating  magnetic  fluxes  the  magnetic  energy  stored  by  the 
current  when  flowing  in  one  direction,  is  in  many  cases  returned  to  the 
circuit  again  when  the  current  is  flowing  in  the  reversed  direction;  it  surges 
to  and  fro  in  the  iron,  being  alternately  positive  and  negative,  like  in  a 
spring  which  is  alternately  compressed  and  released;  hence  calculations 
of  the  amount  of  the  magnetic  power  involved  are  generally  of  no 
importance  in  practice.  Whatever  energy  leaves  the  circuit  is  generally 
in  the  form  of  electrical  energy  (as  in  transformers)  or  heat  (as  in  the 
hysteresis  loss);  in  such  cases  the  power  of  this  energy  in  watts  is  equal 
to  the  amount  in  joules  which  leaves  per  cycle,  multiplied  by  the  frequency. 

There  are  no  specific  units  of  magnetic  power;  such  power  is  usually 
measured  in  watts  or  in  ergs  per  second,  but  it  may  be  measured  in  terms 
of  any  of  the  units  of  power.  The  C.G.S.  unit  is  the  erg  per  second  • 
the  practical  unit  is  the  watt. 

The  relations  to  other  measures  are  as  follows: 
Watts  of  magnetic  power  [W,  w]  =  henrysX final  amperes2 -s-  seconds X 2. 

(For  further  relations  divide  any  of  the  values  given  for  "joules,"  iri  the 
preceding  section,  by  "seconds.") 


PHOTOMETRIC   UNITS. 

The  different  kinds  or  units  or  measures  given  below  and  their  relations 
and  symbols  (except  "cp"  for  candle-power)  are  those  adopted  by  the 
unofficial  International  Congress  at  Geneva  in  1896. 


INTENSITY    OF    LIGHT  [I];     CANDLE    POWER   [cp]. 
(Flux  of  light  -j-  solid  angle;  powers  solid  angle.) 


m  . 

same  total  quantity  or  flux  of  light  radiating  from  a  source,  the  intensity 
in  any  one  direction  becomes  less  as  the  solid  angle  through  which  it  is 
radiated  becomes  greater,  hence  the  intensity  is  the  total  flux  (in  lumens) 
divided  by  the  number  of  units  of  solid  angle  (see  under  the  table  of'units 
of  Solid  Angles,  p.  89). 


these,  an 

use 

s 


;nese,  ana  tne  one  wnicn  is  oemg  generally  acceptea  ana  is  coming  into 
ise  internationally,  is  the  heftier  unit,  an  amyl  acetate  lamp  of  fixed  dimen- 
sions and  height  of  flame.  It  has  been  carefully  investigated  by  the 


INTENSITY   OF   LIGHT. — CANDLE   POWER.        145 

Reichsanstalt  and  its  coefficients  have  been  determined ;  it  was  recom- 
mended for  international  adoption  to  the  International  Electrical  Congress 
at  Chicago  in  1893,  but  was  not  adopted  for  reasons  which  probably  do 
not  exist  now ;  it  has  been  adopted  by  the  Reichsanstalt ;  it  has  also  been 
accepted  tentatively  by  the  National  Bureau  oi  Standards  through  incan- 
descent lamp  secondary  standards  measured  in  terms  of  it  at  the  Reichs- 
anstalt; it  is  endorsed  by  the  American  Institute  of  Electrical  Engineers. 

The  hefner  standard  lamp  is  very  fully  described  with  diagrams  in  an 
article  emanating  from  the  Reichsanstalt,  published  in  the  Zeitschrift  fur 
Instrumentenkunde,  Vol.  XlII,  July,  1893,  p.  257.  In  this  article  the 
hefner  unit  is  denned  as  the  amount  of  light  from  a  flame  burning  free  in 
stationary  pure  air,  from  a  thick  wick  saturated  with  amyl  acetate,  the 
wick  completely  filling  a  circular  wick-tube  of  German  silver,  the  inner 
diameter  of  the  tube  being  8  mm,  and  the  outer  diameter  8.3  mm,  the  free 
length  of  this  tube  being  25  mm ;  the  height  of  flame  is  40  mm  from  the 
edge  of  the  wick-tube,  measured  at  least  10  minutes  after  lighting. 

The  relation  "1  hefner  =  0.88  British  standard  candle"  is  the  one  gen- 
erally used;  it  is  the  relation  adopted  by  the  Reichsanstalt  and  is  the  one 
used  by  the  National  Bureau  of  Standards. 

The  next  most  important  standard  is  the  British  standard  candle  or 
English  spermaceti  candle  or  simply  English  candle,  which  is  about 
13^%  greater  than  the  hefner.  The  definition  of  the  English  candle  used 
by  the  National  Bureau  of  Standards  is  the  relation  1  hefner  =  0.88  English 
candles.  This  standard  is  the  one  universally  referred  to  in  this  country  by 
the  large  incandescent  electric  lamp  manufacturers  and  gas  companies, 
under  the  term  "  candle  power."  Thecand'e  itself  is  not  as  easy  to  use 
nor  as  reliable  or  constant  as  the  hefner,  for  which  reason  the  hefner  is 
generally  used  as  the  ultimate  standard,  the  results  being  subsequently 
reduced  to  such  English  candles  by  the  relation  1  hefner  =  0.88  of  these 
candles.  An  official  specification  of  the  British  standard  candle  is  given 
very  fully  in  an  article  in  the  American  Gas  Light  Journal,  1894,  Jan.  8th, 
p.  41.  The  candle  has  a  special  wick,  is  made  of  spermaceti  mixed  with 
3%  to  4J^%  of  bleached  beeswax,  weighs  about  a  sixth  of  a  pound,  and 
must  burn  at  a  rate  not  greater  than  126  or  less  than  114  grains  per  hour. 
In  the  comparisons  made  by  the  Reichsanstalt  with  the  hefner,  the  height 
of  the  flame  of  this  English  standard  candle  was  45  mm. 

The  unit  called  the  platinum  standard  of  light,  sometimes  im- 
properly called  an  absolute  unit,  is  the  light  emitted  perpendicularly 
from  a  square  centimeter  of  surface  of  melted  platinum  at  the  tempera- 
ture of  its  solidification.  It  is  often  called  the  violle.  This  was  virtually 
adopted  by  an  International  Congress,  but  never  came  into  use,  and  seems 
to  have  been  abandoned.  Its  value  is  not  known  definitely,  but  is  ap- 
proximately 20  candles.  The  bougie  decimale,  sometimes  called  a  pyr,  is 
one  twentieth  of  this,  and  at  the  unofficial  Geneva  Congress  of  1896  its 
value  was  provisionally  considered  to  be  represented  in  practice  by  one 
hefner.  This  is  the  value  which  will  be  used  in  the  following  tables. 

The  German  paraffin  candle  has  gone  out  of  use,  being  replaced  by 
the  hefner. 

The  carcel  is  an  oil  lamp  formerly  largely  used  as  a  standard  in  France; 
the  oil  is  kept  at  a  fixed  level  by  means  of  a  pump  driven  by  clockwork. 

The  Harcourt  pentane  lamp  is  a  flame  using  pentane  gas;  it  is  generally 
made  for  1  and  for  10  candle-power ;  it  is  used  extensively  as  a  secondary- 
standard  and  as  such  seems  to  be  satisfactory,  but  each  lamp  must  be 
calibrated  by  comparison  with  some  standard,  as  it  seems  they  cannot 
be  made  sufficiently  uniform  to  have  a  definite  value  like  the  hefner. 

Standard  incandescent  electric  lamps  which  have  been  very  care- 
fully standardized  by  means  of  the  hefner  lamp  as  the  ultimate  standard, 
can  now  be  purchased  for  use  as  standards.  When  their  voltage  is  care- 
fully adjusted,  which  can  be  done  with  great  accuracy,  they  form  very 
satisfactory  standards  and  are  said  to  be  very  reliable.  They  are  used 
quite  extensively  and  are  probably  the  best  form  of  secondary  standards 
when  a  constant  source  of  electric  current  is  available. 

Probably  the  best  collection  of  values  of  the  various  standards  is  that 
in  a  paper  by  Dr.  Bunte  in  "The  Technical  Standards  of  Light,"  read  before 
the  International  Photometry  Committee  in  June,  1903.  It  is  translated 
in  the  Journal  of  Gas-lighting,  June  30,  1903,  also  in  the  Progressive  Age, 
Sept.  1  aod  15,  1903.  In  the  following  table  those  values  for  which  the 
authority  is  given  as  Bunte,  have  been  taken  from  this  paper. 


146        INTENSITY    OF    LIGHT. CANDLE    POWER. 

The  fallowing  values,  which  are  believed  to  be  the  best  obtainable ,  are 
mostly  only  approximate;  different  authorities  do  not  agree. 

**  Means  accepted  by  the  Ileichsanstalt  and   the   National  Bureau  of 
Standards.     The  names  in  parenthesis  are  the  authorities. 
1  hefiier  =         1.  bougie  decimale.     (Geneva  Congress.) 
=  1.026    bougies  decimales.     (Violle.) 
=  0.89      bougie  decimale.     (Bunte.) 
=  0885   bougie  decimale.     (Laporte.) 
=  0.833    German  candle.     (Bunte.) 
=  O.88**  British  or  English  standard  candle  (universally  accepted 

value).     Aprx.  %. 
=  0.092      carcel      (Bunte.) 
1  bougie  decimale : 

=        1  hefner.     (Geneva  Congress.) 
=   1.13hefners.     (Violle.) 

=  0.88  British  or  English  standard  candle.     (1  hefner  =-0.88  candle.) 
=  0.99  British  or  English  standard  candle.     (Violle.) 
=  0.94  German  candle.     (Violle.) 
=  005  violle.     (Official.) 
1  Py|<  =  l  bougie  decimale. 
1  British  or  .English  standard  candle  [cp]: 

=  1.136  36  hefners  (from  the  accepted  value  of  the  hefner).     Aprx.  %. 
=         1.14  hefners.     (Bunte.) 
=         1.01  bougies  de  imales.     (Bunte.) 
=       0.950  German  candle.     (Bunte.) 
=       0.105  carcel.     (Bunte.) 
1  candle  or  candle  power  [cp]: 

=  in  this  country  and  England  1  British  or  English  standard  candle. 
1  German  paraffin  candle  (20  mrn  diam.): 

=  1.224  hefners.     (German  Gas  and  Water  Committee.) 
=    1.20  hefners.     (Bunte.) 
=    1.16  hefners.    .(Lummer  arid  Brodhun.) 
=    1.05  English  candles.     (Bunte.) 
=    1.07  bougies  decimales. 
=    1.05  bougies  d''rimales.     (Laporte.) 
=  0.110  carcel.     (Bunte.) 
1  carcel  =  10.87  hefners      (Bunte.) 
"         =  10.9    hefners. 

=   9.62  bougies  decimales.     (Violle.) 
=-   9.53  English  candles.     (Bunte.) 
••         =   9.05  German  candles.     (Bunte.) 

=  0.481  violle.     (Violle.) 
1  ri  olle  =  22 .6  hefners.     (  Violle . ) 

=  2O.  bougies  ddcimales.     (Official.) 
"        =    20.  hefners.     (Geneva  Congress.) 
"        =  19.8  English  candles.     (Violle.) 
"        =18.8  German  candles.     (Violle.) 

=  2.08carcels.     (Violle.) 

1  platinum  standard  =  1  violle.     (Official.) 
1  absolute  unit  =  1  violle. 
1  Harcourt  peiitane  lamp: 

=  secondary  standard  made  for  various  candle-pow«rs. 
In  the  following  relations  a  candle  means  a  hefner  unit. 
Can  dies  =  lumens -4- units  solid  an«;le. 

=  luxes  X  (distance  in  meters)2. 
"         =  units  of  brightness  Xsq.  centimeters. 
"         =  lumen-hours  -f-  (units  solid  angle  X  hoirs). 


FLUX  OF   LIGHT.  147 


FLUX  OF  LIGHT  [$] ;  SPHERICAL  OR  HEMI- 
SPHERICAL CANDLE  POWER.  (Candle  powerX 
solid  angle;  power.) 

This  quantity  measures  the  whole  radiation  or  whole  beam  of  light, 
and  is  therefore  equal  to  the  intensity  in  candle  powers  multiplied  by  the 
number  of  units  cf  s  Aid  angle  through  which  the  conical  beam  radiates. 
In  practice  the  solid  angles  more  usually  used  are  either  the  hemisphere 
(=6.283  19  units)  or  the  whole  sphere  (  =  12.566  4  units).  This  quantity, 
being  a  rad  ation  of  energy,  is  of  the  same  nature  as  power,  and  a  relation 
between  the  two  should  exist  and  would  be  called  the  mechanical  equiva- 
lent of. light  but  it  is  not  yet  known;  it  is  believed  to  be  of  the  order  of 
about  5.3  spherical  candles  per  watt  or  0.188  watts  per  spherical  candle. f 

The  unit  of  flux  is  the  amount  of  flux  of  light  in  a  beam  of  one  unit 
solid  angle  (one  which  subtends  a  square  centimeter  at  a  radius  of  one 
centimeter)  in  which  the  intensity  is  one  candle  power.  The  name  of 
this  unit  adopted  by  the  Geneva  Congress  is  lumen.  In  practice  the  units 
spherical  candle  power  and  hemispherical  candle  power  are  often 
used  instead,  referring  in  this  country  and  England  to  the  English  candle. 
In  each  of  the  following  conversion  factors  the  flux  is  the  same,  but 
the  solid  angle  within  which  it  is  confined  is  different.  Aprx.  means 
within  2%. 
1  lumen  =  1.  solid  angle  hefner. 

=  0.159  155  hemispherical  Irfner.    Aprx.  16-^100. 
=  0.140  036  hemispherical  (English)  candle  power.    Aprx.  ^r. 
=  0.079  .777  spherical  hefner.    Aprx.  8^  100. 
=  0.070  P28  spherical  (English)  candle  power.    Aprx.  7  -4- 100. 
1  hemispherical  hefner: 

=   6.283  19  lumens.    Aprx.  614. 

=  0.88  hemispherical  (English)  candle  power.    Aprx.  J£. 

=  0.500  000  spherical  hefner. 

=  0.440  OtX)  spherical   (English)  candle  power.    Aprx.  %. 
1  hemispherical  (English)  candle  power: 

=  7.13999  lumens.     Aprx.  50/r.    - 

=    1.138  36  hemispherical  hefners.    Aprx.  %. 

=  0.568  181  spherical  hefner.    Aprx.  #. 

=  0.500000  spherical  (English)  candle  power. 
I  spherical  hefner: 

=  12.566  4  lumens.     Aprx.  H X  100. 

=         1.76  hemispherical  (English)  candle  powers.    Aprx.  %• 

=  2  hemispherical  hefners. 

=         0.88  spherical  (English)  candle  power.    Aprx.  %. 
\  spherical  (English)  candle  power: 

=  14.280  0  lumens.     Aprx.  #  X  100. 

=  2.272  73  hemispherical  hefners.    Aprx.  9/4. 

=  2.  hemispherical  (English)  candle  powers. 

=  1.13636  spherical  hefners.    Aprx.  %. 

In  the  following  relations  one  candle  means  a  hefner  unit 
Lumen*  =  candles  X  units  solid  angle. 

= candles  Xsq.    meters  of  illuminated  surf  ace -r- (distance  in  me- 
ters)2. 

=  luxes  X  surf  ace  in  sq.  meters. 

=  units  of  brightness  X  surface  in  sq.  cm  X  units  solid  angle. 
=  lumen-hours  -*-  hours. 

t  See  Mechanical  Equivalent  of  Light.  Elec.  World  and  Eng.t  April  20, 
1901, p  631. 


148 


ILLUMINATION . — BRIGHI  NESS. 


ILLUMINATION    [E].      (Candle  power -f- distance2;   flux   of 
light  -f-  surface.) 

This  quantity  measures  the  amount  of  light  falling  on  a  surface;  it 
measures  that  which  is  received  as  distinguished  from  that  which  is  given 
put  by  the  source.  It  is  a  very  important  quantity  in  photometry,  as 
illumination  is  that  which  light  is  intended  to  produce.  The  illumination 
of  a  surface  is  proportional  to  the  candle  power  of  the  source  of  light  and 
inversely  proportional  to  the  square  of  the  distance  of  the  illuminated 
surface  from  the  :  ource ;  if  the  intensity  of  the  source  is  in  hefners  and  the 
t'istance  in  meters,  then  the  illumination  will  be  in  luxes.  The  illumina- 
tion in  luxes  is  also  equal  to  the  total  flux  of  light  in  lumens  shining  on 
the  surface,  divided  by  the  amount  of  the  surface  in  square  meters.  These 
relations  give  the  amount  of  illumination  which  reaches  that  surface  from 
the  source,  and  not  the  amount  of  light  reflected  from  the  surface,  as  that 
depends  on  the  nature  and  color  of  the  surface. 

The  unit  adopted  by  the  Geneva  Congress  is  a  lux,  which  is  equal  to 
the  illumination  produced  by  one  hefner  at  a  distance  of  one  meter.  When 
the  source  is  called  a  c-mdle,  then  this  unit  is  often  called  a  meter-candle  ; 
but  it  is  then  improperly  named,  as  it  should  be  called  a  candle  per  meter 
or  still  more  correctly  a  candle  per  meter  squared.  The  lux  is  also 
equal  to  one  lumen  of  flux  per  square  meter.  A  unit  called  the  foot- 
candle  (more  properly  candle  per  loot  or  candle  per  foot  squared) 
is  also  used ;  it  is  equal  to  the  illumination  produced  by  an  English  candle 
at  a  distance  of  one  foot.  With  these  units  the  amount  of  light  required 
to  produce  any  desired  illumination  at  a  given  distance  can  readily  be 
calculated. 
1  lux  : 

1.  lumen  per  sq.  meter. 

1.  meter-candle  (hefner)  or  1  candle  (hefner)  per  meter  squared. 
=  0.081  8  foot-candle  (English)  or  candle  (English)  per  foot  squared. 

Aprx.  %2-- 
1  meter-candle  (hefner): 

1.  lux. 

=  1.  lumen  per  sq.  meter. 

=  0.081  8  foot-candle  (English).    Aprx.  Vis. 
1  f oo  t-csi  iicllo  (English)  =  12. 2  luxes. 

=  12.2  meter-candles  (hefner). 
=  12.2  lumens  per  sq.  meter. 

In  the  following  relations  a  candle  means  a  hefner  unit. 
Luxe  s  =  candles -s-  (distance  in  meters)2. 

=  flux  in  lumens-:- surface  in  sq.  meters. 

=  flux  in  lumens -^-[(distance  in  meters)2  X  units  solid  angle]. 
=  candles  X  units  solid  angle  -r-  surface  in  sq.  meters. 
=  units  of  brightness  X  surface  of  source  in  sq.  cm.  -s-  (dist.  in  meters)2. 
"       =  lumen-hours -s- (hours X  surface  in  sq.  meters). 


BRIGHTNESS  OP  SOURCE  [e].      (Candle  powers  surface 
of  source.) 

This  quantity  measures  the  brightness  of  the  source;  it  is  the  total 
candle  power  of  the  source  divided  by  its  surface  in  sq.  centimeters.  It 
is  seldom  used.  The  unit  is  one  hefner  per  sq.  centimeter;  no 
name  has  been  given  to  it.  As  there  are  no  other  units  of  this  kind,  there 
are  no  conversion  factors. 
1  unit  of  brightness  =  1  candle  (hefner)  per  sq.  centimeter. 

In  the  following  relations  a  candle  means  a  hefner  unit. 
Units  of  brightness  : 

=  candles  -r-  centimeter2. 

=  flux  in  lumens -*-( centimeter2 X  units  solid  angle). 


QUANTITY    OF   LIGHT. — EFFICIENCY.  149 


QUANTITY  OF   LIGHT  [Q],      (Flux  of  lightX  time  ; 
energy.) 

This  quantity  measures  the  total  amount  or  volume  of  light  together 
with  its  duration.  A  rational  payment  for  light,  for  instance,  would 
be  made  in  terms  of  this  unit.  It  is  equal  to  the  amount  of  flux  in  lumens 
multiplied  by  the  time  in  hours. 

The  unit  is  one  lumen  of  flux  for  one  hour,  and  is  called  a  lumen-hour. 
As  there  are  no   other  units  of  the  same  kind,  there  are  no  conversion 
/actors. 
1  lumen- hour=l  lumen  for  one  hour. 

In  the  following  relations  a  candle  means  a  hefner  unit. 
Lumen-hours : 

=  flux  in  lumens  X  hours. 

=  candles  X  hours  X  units  solid  angle. 

«=  candles  X  illuminated    surface    in   sq.   meters X  hours -5- (distance    in 

meters)2. 

=  luxes  X  illuminated  surface  in  sq.  meters  X  hours. 
=  units  of  brightness  X  sq.  cm  of  source  X  units  solid  angle  X  hours. 
=  units  of  brightness  Xsq.  cm  of  source  X  illuminated   surface    in  sq. 

meters  X  hours  •*•  (distance  in  meters)2. 


LIGHT   EFFICIENCY;    POWER   PER   CANDLE 
POWER.     ( Candles ~ power;  power ~ candles.) 

The  efficiencies  of  electric  lights  are  frequently  compared  with  each 
other  by  comparing  the  watts  required  per  candle.  The  number  thus 
obtained,  by  dividing  the  watts  by  the  candles,  is  sometimes  called  the 
efficiency,  which  term,  however,  is  incorrectly  used,  because  th:  greater 
the  watts  per  candle,  the  lower  the  efficiency;  the  term  efficiency  would  be 
more  correctly  applied  to  the  number  giving  the  candles  per  watt.  Com- 
parisons between  such  relative  efficiencies  are  very  useful,  even  though  the 
figures  are  not  the  absolute  efficiencies;  for  the  latter  it  would  be  necessary 
to  know  the  mechanical  equivalent  of  light ,t  and  instead  of  the  candle 
power,  the  total  flux  in  lumens  or  the  soherical  candle  powers  should  be 
used,  so  as  to  include  the  solid  angle  through  which  the  light  is  radiated. 

The  following  relations  apply  equally  well  to  hefners  as  to  English 
candles. 

n  watts  per  candle  =  l/n  candles  per  watt. 
n  candles  per  watt  =  l/n  watts  per  candle. 

n  watts  per  candle=735.448/n  candles  per  metric  horse-power. 
'*  =745.650/n  candles  per  horse-power. 

=      1  000/tt  candles  per  kilowatt. 

n  candles  per  metric  horse-power  =735. 448/n  watts  per  candle. 
n  candles  per  horse-power  =745. 650/n  watts  per  candle. 
n  candles  per  kilo  watt  =  1  000 /n  watts  per  candle. 

For  the  relation  between  watts,  horse-powers,  etc.,  see  table  of  units  of 
Power,  page  80. 

t  S«e  notes  on  Flux  of  Light,  p.  147,  and  the  foot-note. 


150  THERMOMETER    SCALES. 


THERMOMETER    SCALES. 

Of  the  four  scales  in  use,  the  Centigrade  scale  (also  called  Celsius)  is  the 
most  rational  one  and  the  one  used  in  all  scientific  research  and  interna- 
tional literature;  it  is  also  used  exclusively  in  some  of  the  European  coun- 
tries. The  zero  point  is  the  melting  point  of  ice,  and  the  100°  point  is  the 
boiling  point  of  water.  The  Fahrenheit  scale  is  used  in  the  United  States 
and  England ;  on  this  scale  the  melting  point  of  ice  is  exactly  32°  and  the 
boiling  point  of  water  is  212°.  The  Reaumur  scale  is  in  limited  use  in  Ger- 
many; it  has  the  same  zero^  point  as  the  Centigrade  scale,  but  the  boiling 
point  of  water  on  this  scale  is  exactly  80°.  The  Absolute  scale  begins  at  a 
theoretical,  assumed  point,  supposed  at  present  to  be  the  lowest  tempera- 
ture which  can  exist ;  this  point  is  calculated  from  the  expansion  of  gases 
at  ordinary  temperatures  and  it  is  assumed  that  the  same  law  holds  good 
down  to  an  absolute  zero ;  it  has  never  been  reached,  but  has  been  approached 
to  within  17  degrees  of  the  Centigrade  scale. 

The  following  table  gives  the  corresponding  values  on  these  four  different 
scales  for  the  complete  range  of  all  known  temperatures,  thus  avoiding 
most  of  the  reductions.  To  these  has  been  added  a  "concrete  scale," 
which  gives  numerous  temperatures  at  which  certain  materials  change  some 
property,  thereby  enabling  one  to  establish,  maintain,  or  measure  those 
temperatures.  But  as  most  of  these  temperatures  are  not  known  defi- 
nitely, they  cannot  be  relied  upon  for  more  than  approximate  correctness. 
They  have  been  compiled  from  a  large  number  of  sources  with  due  regard 
for  the  authorities;  many  of  them  were  taken  from  Carnelley's  excellent 
table  of  melting  and  boiling  points,  and  from  Landolt  and  Boernstein's 
tables. 

Reduction  factors  for  one  degree: 

A  Centigrade  degree  is  %  or  1.8  Fahrenheit  degrees.  It  is  %  or  0.8 
Rdaumur  degree.  It  is  the  same  as  a  degree  of  the  Absolute  scale. 

A  Fahrenheit  degree  is  %  or  a  little  more  than  half  of  a  Centigrade 
degree  or  of  a  degree  of  the  Absolute  scale.  It  is  %  of  a  Reaumur  degree. 

A  Keaumur  degree  is  %  or  1.25  Centigrade  degrees,  or  degrees  of  the 
Absolute  scale.  It  is  %  or  2.25  Fahrenheit  degrees. 

A  degree  of  the  Absolute  scale  is  the  same  as  of  the  Centigrade  scale. 

Reduction  factors  for  readings  of  a  temperature  in  degrees; 

To  convert  a  reading  in  Centigrade  degrees  into  the  corresponding 
one  in  Fahrenheit  degrees,  multiply  by  %  and  add  32.  To  convert  it  into 
the  one  in  Reaumur  degrees  multiply  by  %.  To  convert  it  into  the  one  on 
the  Absolute  scale,  add  273. 

To  convert  a  reading  in  Fahrenheit  degrees  into  the  one  in  Centi- 
grade degrees,  subtract  32  and  then  multiply  by  %,  being  careful  about  the 
signs  when  the  reading  is  below  the  melting  point  of  ice.  To  convert  it 
into  the  one  in  Reaumur  degrees,  subtract  32  and  multiply  by  %,  To  con- 
vert it  into  the  one  on  the  Absolute  scale,  subtract  32,  then  multiply  by  % 
and  add  273;  or  multiply  by  5.  add  2  297.  and  divide  by  9. 

To  convert  a  reading  in  Reaumur  degrees  into  the  one  in  Centigrade 
degrees,  multiply  by  %.  To  convert  it  into  the  one  in  Fahrenheit  degrees, 
multiply  by  %  and  add  32.  To  convert  it  into  the  one  on  the  Absolute  scale, 
multiply  by  %  and  add  273. 

To  convert  a  reading  on  the  Absolute  scale  to  the  one  in  Centigrade 
degrees,  subtract  273.  To  convert  it  into  the  one  in  Fahrenheit  degrees  sub- 
tract 273,  multiply  by  %,  and  add  32;  or  multiply  by  9,  subtract  2297.  and 
divide  by  5.  To  convert  it  into  the  one  in  Reaumur  degrees  subtract  273 
and  multiply  by  %. 

All  these  reduction  factors  are  strictly  correct.  Care  must  be  taken  to 
add  or  subtract  algebraically  when  any  of  the  readings  are  belov  the  zero 
of  the  respective  scale,  in  which  case  they  should  be  preceded  by  the  nega- 
tive sign.  There  can  be  no  negative  values  on  the  Absolute  scale  (unless  it 
is  shown  in  the  future  that  the  absolute  zero  has  been  fixed  too  high,  a  pos- 
sibility which  may  not  be  remote). 


THERMOMETER   SCALES. 


151 


THERMOMETER  SCALES. 


Centi- 

Fahren- 

Re'au- 

Abso- 

grade 

heit 

mur 

lute 

Concrete  Scale  (mostly  only  approximate). 

Deg. 

Deg. 

Deg. 

Deg. 

-273 

-459.4 

-218.4 

0 

-  256°  C.  hydrogen  freezes 

-250 

-418.0 

-200.0 

+   23 

-250°  C.  hydrogen  boils 

-225 

-373.0 

-180.0 

+   48 

—  214°  C.  nitrogen  freezes 

-200 

-328.0 

-160.0 

+   73 

—  200°  C.  temperature  of  liquid  air 

-190 

-310.0 

-152.0 

+   83 

-  193.1°  to  -  194°  C.  (760  mm  press.)  nitro- 

[gen  boils 

-180 

-292.0 

-144.0 

+   93 

-  181.4°  to  -  184°  C.  (1  atm.)  oxygen  boils 

-170 

-274.0 

-136.0 

+  103 

—  167.0°  C.  nitric  oxide  solidifies 

-160 

-256.0 

-128.0 

+  113 

-150 

-238.0 

-120.0 

+  123 

-  153.6°  C.  nitric  oxide  boils 

-140 

-220.0 

-112.0 

+  133 

-130 

-202.0 

-104.0 

+  143 

-  130.5°  C.  pure  ethyl  alcohol  freezes 

-120 

-184.0 

-   96.0 

+  153 

-110 

-166.0 

-   88.0 

+  163 

-  1  12.5°  C.  hydrochloric  acid  melts 

-100 
-  90 

-148.0 
-130.0 

-   80.0 
-    72.0 

+  173 

+  183 

—  102°  C.  hydrochloric  acid  condenses 
—  85.5°  C.  hydrogen  sulphide  solidifies 

-   80 

-112.0 

-   64.0 

+  193 

—  76°  C.  sulphur  dioxide  solidifies 

-   70 

-   94.0 

-   56.0 

+  203 

—  75°  C.  ammonia  melts 

-  60 

-   76.0 

—   48.0 

+  213 

—  61.8°  C.  hydrogen  sulphide  boils  at  760 

-   50 

-   58.0 

-   40.0 

+  223 

[mm 

-   45 

-   49.0 

-   36.0 

+  228 

[sol.)  solidifies 

-   40 

-   40.0 

-   32.0 

+  233 

-38°  to  -41°  C.  ammonium  hydrate  (sat. 

-   35 

-   31.0 

-    28.0 

+  238 

—  39.5°  C.  mercury  melts 

-   30 

-   22.0 

-    24.0 

+  243 

—  33.7°  C.  ammonia  boils 

-   28 

-    18.4 

-   22.4 

+  245 

-  33.6°  C.  chlorine  boils 

-  26 

-    14.8 

-   20.8 

+  247 

-  24 

-    11.2 

-    19.2 

+  249 

-   22 

-     7.6 

-   17.6 

+  251 

-   20 

-     4.0 

-    16.0 

+  253 

-  20.7°  C.  (CN)2  boils 

—    19 

-     2.2 

-    15.2 

+  254 

-    18 

-      0.4 

-    14.4 

+  255 

-  18.0°  C.SbH3  boils 

-17.8 

0 

-14.2 

+  255.2 

-17.2 

+      1.0 

-    13.8 

+  255.8 

-17.0 

+      1.4 

-    13.6 

+  256.0 

-16.7 

+      2.0 

-    13.3 

+  256.3 

-16.1 

+     3.0 

-    12.9 

+  256.9 

-16.0 

+     3.2 

-    12.8 

+  257.0 

-15.6 

+      4.0 

-    12.4 

+  257.4 

-15.0 

+      5.0 

-    12.0 

+  258.0 

-14.4 

+      6.0 

-    11.6 

+  258.6 

-14.0 

+      6.8 

-    11.2 

+  259.0 

-13.9 

+     7.0 

-    11.1 

+  259.1 

-13.3 

+      8.0 

-    10.7 

+  259.7 

-13.0 

+      8.6 

-    10.4 

+  260.0 

-12.8 

+      9.0 

-    10.2 

+  260.2 

-12.2 

+    10.0 

-      9.8 

+  260.8 

-12.0 

+    10.4 

-      9.6 

+  261.0 

-11.7 

+    11.0 

-      9.3 

+  261.3 

-11   1 

+    12.0 

-      8.9 

+  261.9 

-11.0 

+    12.2 

-      8.8 

+  262.0 

-10.6 

+    13.0 

-      8.4 

+  262.4 

-  10.5°  C.  sulphur  dioxide  boils  at  744  mm 

152 


THERMOMETER    SCALES. 


THERMOMETER  SCALES— (Continued} 


Centi- 
grade 
Deg. 

Fahren- 
heit 
Deg. 

R^au- 
mur 
Deg. 

Abso- 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate). 

-10.0 

+  14.0 

-8.0 

+  263.0 

-   9.4 

+  15.0 

-7.6 

+  263.6 

-   9.0 

+  15.8 

-7.2 

+  264.0 

-   8.9 

+  16.0 

-7.1 

+  264.1 

-   8.3 

+  17.0 

-6.7 

+  264.7 

—  8.5°  C.  sulphuric  acid  melts,  sp.  gr 

.  1.732 

-   8.0 

+  17.6 

-6.4 

+  265.0 

-   7.8 
-   7.2 

+  18.0 
+  19.0 

-6.2 

-5.8 

+  265.2 
+  265.8 

—  7.5°  C.  sulphuric  acid  melts,  sp.  gr 
—  7.2°  C.  bromine  solidifies 

.  1.727 

-   7.0  +19.4 
-   6.71+20.0 

-5.6 
-5.3 

+  266.0 
+  260.  3 

-  6.1 

+  21.0 

-4.9 

+  266.9 

-   6.0 

+  21.2 

-4.8 

+  267.0 

-   5.6 

+  22.0 

-4.4 

+  267.4 

-   5.0 

+  23.0 

-4.0 

+  268.0 

—   4.4 

+  24.0 

-3.6 

+  268.6 

-   4.0 

+  24.8 

-3.2 

+  269.0 

-   3.9 

+  25.0 

-3.1 

+  269.1 

-   3.3 

+  26.0 

-2.7 

+  269.7 

-   3.0 

+  26.6 

-2.4 

+  270.0 

' 

-   2.8 

+  27.0 

-2.2 

+  270.2 

-    2.2 

+  28.0 

-1.8 

+  270.8 

-   2.0 

+  28.4 

-1.6 

+  271.0 

-    1.7 

+  29.0 

-1.3 

+  271.3 

-    1.1 

+  30.0 

-0.9 

+  271.9 

-    1.0 

+  30.2 

-0.8 

+  272.0 

-   0.6 

+  31.0 

-0.4 

+  272.4 

0 

+  32.0 

0 

+  273.0 

0°  C.  freezing  point  of  water 

+  0.6 

+  33.0 

+  0.4 

+  273.6 

1.0 

33.8 

0.8 

274.0 

1.1 

34.0 

0.9 

274.1 

1.7 

35.0 

1.3 

274.7 

2.0 

35.6 

1.6 

275.0 

2.2 

36.0 

1.8 

275.2 

2.8 

37.0 

2.2 

275.8 

3.0 

37.4 

2.4 

276.0 

3°  C.  benzene  (benzol  )  freezes 

3.3 

38.0 

2.7 

276.3 

3.9 

39.0 

3.1 

276.9 

4.0 

39.2 

3.2 

277.0 

[molecular  proportion 

me'ts 

4.4 

40.0 

3.6 

277.4 

4.5°  C.  alloy  of  potassium  and  sodi 

um  in 

5.0 

41.0 

4.0 

278.0 

4.5*  C.  sulphuric  acid  melts,  sp.  gr.  1 

.749 

5.6 

42.0 

4.4 

278.6 

6.0 

42.8 

4.8 

279.0 

6°  C.  alloy  of  1  potassium  and  1  sodium 

6.1 

43.0 

4.9 

279.1 

[melts 

6.7 

44.0 

5.3 

279.7 

7.0 

44.6 

5.6 

280.0 

7.2 

45.0 

5.8 

280.2 

7.8 

46.0 

6.2 

280.8 

8.0 

46.4 

6.4 

281.0 

8.3 

47.0 

6.7 

281.3 

8.9 

48.0 

7.1 

281.9 

THERMOMETER    SCALES. 


153 


THERMOMETER  SCALES— (Continued). 


Centi- 
grade 
Deg. 

Fahren- 
heit 
Deg. 

R&m- 
mur 
Deg. 

Abso- 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate). 

9.0 

48.2 

7.2 

282.0 

9.4 

49.0 

7.6 

282.4 

10.0 

50.0 

8.0 

283.0 

10.6 

51.0 

8.4 

283.6 

10.5°  C.  pure  sulphuric  acid  freezes, 

sp  .  gr. 

11.0 

51.8 

8.8 

284.0 

[1.854 

11.1 

52.0 

8.9 

284.1 

11.7 

53.0 

9.3 

284.7 

12.0 

53.6 

9.6 

285.0 

12.2 

54.0 

9.8 

285.2 

12.8 

55.0 

10.2 

285.8 

13.0 

55.4 

10.4 

286.0 

13.3 

56.0 

10.7 

286.3 

13.9 

57.0 

11.  1 

286.9 

14.0 

57.2 

11.2 

287.0 

14.4 

58.0 

11.6 

287.4 

15.0 

59.0 

12.0 

288.0 

15.6 

60.0 

12.4 

288.6 

16.0 

60.8 

12.8 

2S9.0 

16.1 

61.0 

12.9 

289.1 

16.7 

62.0 

13.3 

289.7 

17.0 

62.6 

13.6 

290.0 

17°  C.  pure  acetic  acid  freezes 

17.2 

63.0 

13.8 

290.2 

17.8 

64.0 

14.2 

290.8 

18.0 

64.4 

14.4 

291.0 

18.3 

65.0 

14.7 

291.3 

18.9 
19.0 

66.0 
66.2 

15.1 
15.2 

291.9 
292.0 

19°-20°  C.  fresh  butter  solidifies 

19  4 

67.0 

15.6 

292.4 

20.0 
20.6 

68.0 
69.0 

16.0 
16.4 

293.0 
293.6 

20°-20.5°  C.  cocoanut  oil  solidifies 
20.5°  C.  cocoa  butter  solidifies 

21.0 

69.8 

16.8 

294.0 

21°  C.  fresh  soft  palm  oil  solidifies 

21.1 

70.0 

16.9 

294.1 

21.7 

71.0 

17.3 

294.7 

22.0 

71.6 

17.6 

295.0 

22.2 

72.0 

17.8 

295.2 

22.8 

73.0 

18.2 

295.8 

23.0   '     73.4 

18.4 

296.0 

23.3         74  0 

18.7 

296.3 

23.9 
24.0 

75.0 
75.2 

19.1 
19.2 

296.9 
297.0 

24°  C.  fresh  hard  palm  oil  solidifies 

24.4 

76  0 

19.6 

297.4 

24.5°  C.  cocoanut  oil  melts 

25.0 

77.0 

20.0 

298.0 

25.6 

78.0 

20.4 

298.6 

26.0 

78.8 

20.8 

299.0 

26.1 

79.0 

20.9 

299.1 

26.7 

80.0 

21.3 

299.7 

26.5°  C.  pure  hydrocyanic  acid  boils 

27.0 

80.6 

21.6 

300.0 

27.2 

81.0 

21.8 

300.2 

27.8 

82.0 

22.2 

300.8 

28.0 

82.4 

22.4 

301.0 

154 


THERMOMETER    SCALES. 


THERMOMETER  SCALES— (Continued). 


Centi- 

Fahren- 

Reau- 

Abso- 

grade 
Deg. 

heit 
Deg. 

mur 
Deg. 

lute 
Deg. 

Concrete  Scale  (mo«tly  only  approximate) 

28.3 

83.0 

22.7 

301.3 

28.5°  C.  calcium  chloride,  (CaCl2  +  6H9O), 

28.9 

84.0 

23.1 

801.9 

[melts  (see  also  under  719°  C.) 

29.0 

84.2 

23.2 

302.0 

29.4 

85.0 

23.6 

302.4 

30°  C.  lard  solidifies 

30.0 

86.0 

24.0 

303.0 

30°  C.  fresh  soft  palm  oil  melts 

30.6 

87.0 

24.4 

303.6 

30°  C.  gallium  melts 

31.0 

87.8 

24.8 

304.0 

31.1 

88.0 

24.9 

304.1 

31°-31.5°  C.  fresh  butter  melta 

31.7 

89.0 

25.3 

304.7 

32.0 

89.6 

25.6 

305.0 

32.2 

90.0 

25.8 

305.2 

32.8 

91.0 

26.2 

305.8 

33.0 

91.4 

26.4 

306.0 

33°  C.  nutmeg  butter  solidifies 

33.3 

92.0 

26.7 

306  .  3 

33°  C.  fresh  beef  tallow  solidifies 

33.9 

93.0 

27.1 

306.9 

33.5°-34°  C.  cocoa  butter  melts 

34.0 

93.2 

27.2 

307.0 

34°  C.  old  beef  tallow  solidifies 

34.4 

94.0 

27.6 

307.4 

34.5°  C.  ether  boils 

35.0 

95.0 

28.0 

308.0 

35.6 

96.0 

28.4 

308,6 

36.0 

96.8 

28.8 

309.0 

36°  C.  fresh  mutton  tallow  solidifies 

36.1 

97.0 

28.9 

309.1 

36.7 

98.0 

29.3 

309.7 

37.0 

98.6 

29.6 

310.0 

37°  C.  blood  heat  of  the  human  body 

37.2 

99.0 

29.8 

310.2 

37.8 

100.0 

30.2 

310.8 

38°  C.  fresh  hard  palm  oil  melts 

38.0 

100.4 

30.4 

311.0 

38°  C.  old  palm  oil  solidifies 

38.3 

101.0 

30.7 

311.3 

38°  C.  pentane  boi!^ 

38.9 

102.0 

31.1 

311.9 

38.5°  C.  rubidium  melt:; 

39.0 

102.2 

31.2 

312.0 

39.5°  C.  old  mutton  tallow  solidifies 

39.4 

103.0 

31.6 

312.4 

39.5°  C.  crystalline  phenol  (carbolic  acid) 

[melts 

40.0 

104.0 

32.0 

313.0 

40°  C.  magnesium  chlorate  melts 

40.6 

105.0 

32.4 

313.6 

40.5°-41°  C.  Japanese  wax  solidifies 

41.0 

105.8 

32.8 

314.0 

41.1 

106.0 

32.9 

314.1 

41.7 

107.0 

33.3 

314.7 

41.5°-42°  C.  lard  melts 

42.0 

107.6 

33.6 

315.0 

42°  C.  old  palm  oil  melts 

42.2 

108.0 

33.8 

315.2 

42.8 

109.0 

34.2 

315.8 

43.0 

109.4 

34.4 

316.0 

43°  C.  fresh  beef  tallow  melts 

43.3 

110.0 

34.7 

316.3 

43.5°  C.  old  beef  tallow  melts 

43.9 

111.0 

35.1 

316.9 

43.5°-44°  C.  nutmeg  butter  melts 

44.0 

111.2 

35.2 

317.0 

44°  C.  spermaceti  solidifies 

44  4 

112.0 

35.6 

317.4 

440-44.5°  C.  spermaceti  melts 

45.0 

113.0 

36.0 

318.0 

44.2°-44.5°  C.  yellow  phosphorus  melts 

45.6 

114.0 

36.4 

318.6 

46.0 

114.8 

36.8 

319.0 

46.1 

115.0 

36.9 

319.1 

46  7 

116.0 

37.3 

319.7 

47^0 

116.6 

37.6 

320.0 

47°  C.  fresh  mutton  tallow  melts 

47.2 

117.0 

37.8 

320.2 

THERMOMETER    SCALES. 


155 


THERMOMETER  SCALES— (Continued). 


Centi- 
grade 
Deg. 

Fahren- 
heit 
Deg. 

R(%u- 
mur 
Deg. 

Abso- 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate). 

47.8 

118.0 

38.2 

320.8 

48.0 

118.4 

38.4 

321.0 

48.3 

119.0 

38.7 

321.3 

48.9 

120.0 

39.1 

321.9 

49.0 

120.2 

39.2 

322.0 

49.4 

121.0 

39.6 

322  .4 

50.0 
50.6 

122.0 
123.0 

40.0 
40.4 

323.0 
323.6 

50°  C.  hydrogen  perchlorate  melts 
50.5°  C.  old  mutton  tallow  melts 

51.0 

123.8 

40.8 

324.0 

51.1 

124.0 

40.9 

324.1 

51.7 

125.0 

41.3 

324.7 

52.0 

12.5  6 

41.6 

325.0 

52.2 

126.0 

41.8 

325.2 

52.8 

127.0 

42.2 

325.8 

53.0 

127.4 

42.4 

326.0 

53.3 

128.0 

42.7 

326  .  3 

53.9 

129.0 

43.1 

326.9 

54.0 

129.2 

43.2 

327.0 

53.5°-54.5°  C.  Japanese  wax  melts 

54  .  4 

130.0 

43.6 

327.4 

55.0 

131.0 

44.0 

328.0 

55°  C.  methyl  alcohol  boils 

55.  G 

132.0 

44.4 

328.6 

50.0 

132.8 

44.8 

329  .  0 

56.1 

133.0 

44.9 

329.1 

56.7 

134.0 

45.3 

329.7 

57.0 

134.6 

45.6 

330.0 

57.2 

135.0 

45.8 

330.2 

57.8 

136.0 

46.2 

330.8 

58.0 

136.4 

46.4 

331.0 

58.3 

137.0 

46.7 

331.3 

58.9 

138.0 

47.1 

331.9 

59.0 

138.2 

47.2 

332.0 

59.4 

139.0 

47.6 

332.4 

60.0 

140.0 

48.0 

333.0 

60.6 

111.0 

48.4 

333.6 

60.5   C.  Wood's  alloy  (BijCdPbzSn)  melts 

61.0 

141.8 

48.8 

334.0 

61.1 

142.0 

48.9 

334.1 

61.7 

143.0 

49.3 

334.7 

62.0 

143.6 

49.6 

335  .  0 

62°  C.  chloroform  boils 

62.2 

144.0 

49.8 

335.2 

62°-62.5°  C.  yellow  bees'  wax  melts 

62.8 

145.0 

50.2 

335.8 

62.1°  C.  potassium  melts 

63.0 

145.4 

50.4 

336.0 

63°  C.  bromine  boils 

63.3 

146.0 

50.7 

336.3 

63°-63.5°  C.  white  bees'  wax  melts 

63.9 

147.0 

51.1 

336.9 

64.0 

147.2 

51.2 

337.0 

64.4 

148.0 

51.6 

337.4 

65.0 

149.0 

52.0 

338.0 

65.6 

150.0 

52.4 

338.6 

65.5°  C.  alloy  Cd4Sn6Pb6Biio  melts 

66.0 

150.8 

52.8 

339.0 

66.1 

151.0 

52.9 

339  .  1 

66.7 

152.0 

53.3 

339.7 

156  THERMOMETER    SCALES. 

THERMOMETER  SCALES— (Continued). 


Centi- 
grade 
Deg. 

Fahren- 
heit 
Deg. 

R(5au- 
mur 
Deg. 

Abso- 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate). 

67.0 

152.6 

53.6 

340.0 

67.2 

153.0 

53.8 

340.2 

67.8 

154.0 

54.2 

340.8 

67.5°  C.  alloy  CdaSn4Pb4Bi8  melts 

68.0 

154.4 

54.4 

341.0 

68.3 

155.0 

54.7 

341.3 

68.9 

156.0 

55.1 

341.9 

68.5°  C.  alloy  CdSnPbBi2  melta 

69.0 

156.2 

55.2 

342.0 

69.4 

157.0 

55.6 

342.4 

70.0 

158.0 

56.0 

343.0 

70°  C.  hexane  boils 

70.6 

159.0 

56.4 

343.6 

71.0 

159.8 

56.8 

344.0 

71.1 

160.0 

56.9 

344.1 

71.7 

161.0 

57.3 

344.7 

72.0 

161.6 

57.6 

345.0 

72.2 

162.0 

57.8 

345.2 

72.8 

163.0 

58.2 

345.8 

73.0 

163.4 

58.4 

346.0 

73.3 

164.0 

58.7 

346.3 

73.9 

165.0 

59.1 

346.9 

74.0 

165.2 

59.2 

347.0 

74.4 

166.0 

59.6 

347.4 

75.0 

167.0 

60.0 

348.0 

75.6 

168.0 

60.4 

348.6 

76.0 

168.8 

60.8 

349.0 

V6.1 

169.0 

60.9 

349.1 

76.7 

170.0 

61.3 

349.7 

77.0 

170.6 

61.6 

350.0 

77.2 

171.0 

61.8 

350.2 

77.8 

172.0 

62.2 

350.8 

78.0 

172.4 

62.4 

351.0 

78.3 

173.0 

62.7 

351.3 

78.4°  C.  pure  ethyl  alcohol  boils 

78.9 

174.0 

63.1 

351.9 

79.0 

174.2 

63.2 

352.0 

79.4 

175.0 

63.6 

352.4 

79.2°  C.  naphthalene  melts 

80.0 

176.0 

64.0 

353.0 

80.6 

177.0 

64.4 

353.6 

80.4°  C.  benzene,  at  760  mm  pressure,  boils 

81.0 

177.8 

64.8 

354.0 

81.1 

178.0 

64.9 

354.1 

81.7 

179.0 

65.3 

354.7 

82.0 

179.6 

65.6 

355.0 

82.2 

180.0 

65.8 

355.2 

82.8 

181.0 

66.2 

355.8 

83.0 

181.4 

66.4 

356.0 

83  .  3 

182.0 

66.7 

356.3 

83.9 

183.0 

67.1 

356.9 

84.0 

183.2 

67.2 

357.0 

84.4 

184.0 

67.6 

357.4 

85.0 

185.0 

68.0 

358.0 

85.6 

186.0 

68.4 

358  .  6 

86.0 

186.8 

68.8 

359.0 

THERMOMETER    SCALES. 


157 


THERMOMETER  SCALES  (Continued). 


Centi- 
grade 
Deg. 

Fahren- 
heit 
Deg. 

Rdau- 
mur 
Deg. 

Abso- 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate). 

86.1 

187.0 

68.9 

359.1 

86.7 

188.0 

69.3 

359.7 

87.0 

188.6 

69.6 

360.0 

87.2 

189.0 

69.8 

360.2 

87.8 

190.0 

70.2 

360.8 

88.0 

190.4 

70.4 

361.0 

88.3 

191.0 

70.7 

361.3 

88.9 

192.0 

71.1 

361.9 

89.0 

192.2 

71.2 

362.0 

89.4 

193.0 

71.6 

362.4 

89.5° 

C.  alloy  CdPb3Bi4  melts 

90.0 

194.0 

72.0 

363.0 

90.6 

195.0 

72.4 

363.6 

91.0 

195.8 

72.8 

364.0 

91.1 

196.0 

72.9 

364  1 

91.7 

197.0 

73.3 

364.7 

92.0 

197.6 

73.6 

365.0 

92°  C 

.  potash  alum  melts 

92.2 

198.0 

73.8 

365  2 

92.8 

199.0 

74.2 

365.8 

93.0 

199  4 

74.4 

366.0 

93.3 

200.0 

74.7 

366.3 

93.9 

201.0 

75.1 

366.9 

93.7° 

C.  Rose's  alloy,  Bi2PbSn,  melts 

94.0 

201.2 

75.2 

367  0 

94.4 

202.0 

75.6 

367.4 

95.0 

203  .  0 

76.0 

368.0 

95°  C 

.  alloy  Cd2Pb7Bi8  melts 

95.6 

204.0 

76.4 

368.6 

95.6° 

C.  sodium  melts 

96.0 

204.8 

76.8 

369.0 

96.1 

205.0 

76.9 

369  .  1 

96.7 

206.0 

77.3 

369.7 

97.0 

206.6 

77.6 

370.0 

97.2 

207.0 

77.8 

370.2 

97.8 

208.0 

78.2 

370.8 

98.0 

208.4 

78.4 

371.0 

98.3 

209.0 

78.7 

371.3 

98.9 

210.0 

•  79.1 

371.9 

99.0 

210.2 

79.2 

372.0 

99.4 

211.0 

79.6 

372  .4 

100.0 

212.0 

80.0 

373.0 

100°  C.  water  boils 

100.6 

213.0 

80.4 

373.6 

101.0 

213.8 

80.8 

374.0 

101.1 

214.0 

80.9 

374.1 

101.7 

215.0 

81.3 

374.7 

102.0 

215.6 

81.6 

375.0 

102.2 

216.0 

81.8 

375.2 

• 

102.8 

217.0 

82.2 

375.8 

103.0 

217.4 

82.4 

376.0 

103.3 

218.0 

82.7 

376  .  3 

103.9 

219.0 

83.1 

376.9 

104.0 

219.2 

83.2 

377.0 

104.4 

220.0 

83.6 

377.4 

104.4°  C.  vitreous  selenium  melts 

105.0 

221.0 

84.0 

378.0 

158 


THERMOMETER    SCALES. 


THERMOMETER  SCALES— (Continued). 


Centi- 
grade 
Deg. 

Fahren- 
heit 
Deg. 

Abso- 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate). 

105.6 

222.0 

378.6 

106.0 

222.8 

379.0 

106.1 

223.0 

379.1 

106.7 

224.0 

379.7 

107.0 

224.6 

380.0 

107.2 

225.0 

380.2 

107.8 

226.0 

380.8 

108.0 

226.4 

381.0 

108.3 

227.0 

381.3 

108.9 

228.0 

381.9 

109.0 

228.2 

382.0 

109.4 

229.0 

382.4 

110.0 

230.0 

383.0 

110.6 

231.0 

383.6 

111.0 

231.8 

384.0 

111.1 

232.0 

384  .  1 

111.7 

233.0 

384.7 

112.0 

233.6 

385.0 

112.2 

234.0 

385.2 

112.8 

235.0 

385.8 

110°-115°  C.  ferric  sulphate  melts 

113.0 

235.4 

386.0 

113.3 

236.0 

386.3 

113.9 

237.0 

386.9 

114.0 

237  .  2 

387.0 

114.4 
115.0 

238.0 
239.0 

387.4 
388.0 

114.5°  C.   rhombic   sulphur   and    copper    nitrate. 
[(Cu(N03)23H20),  melt 
11  5°  C.  iodine  melts 

115.6 

240.0 

388.6 

116.0 

240.8 

389.0 

f 

116.1 

241.0 

389  .  1 

116.7 

242.0 

389.7 

117.0 

242.6 

390.0 

117.2 

243.0 

390.2 

117.8 

244.0 

390.8 

118.0 

244.4 

391.0 

118.3 

245.0 

391.3 

118.9 

246.0 

391.9 

119.0 

246.2 

392.0 

119.4 

247.0 

392.4 

120.0 

248.0 

393.0 

120°  C.  prismatic  sulphur  melts 

120.6 

249.0 

393.6 

121.0 

249  .  8 

394.0 

121.1 

250.0 

394.1 

121.7 

251.0 

394.7 

122.0 

251.6 

395.0 

122.2 

252.0 

395.2 

122.8 

253.0 

395.8 

123.0 

253.4 

396.0 

123.3 

254.0 

396.3 

123.9 

255.0 

396.9 

124.0 

255.2 

397.0 

THERMOMETER    SCALES. 


159 


THERMOMETER  SCALES— (Continued}. 


Centi- 
grade 
Deg. 

Fahren 
heit 
Deg. 

Abso- 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate 

124.4 

256.0 

39'.    4 

125.0 

257.0 

398.0 

125.3°  C.  alloy  Pb3Bi8  melts 

125.6 

258.0 

398.6 

126  0 

258.8 

399.0 

126.1 

259.0 

399.1 

126.7 

260.0 

399.7 

127.0 

260.6 

400.0 

127.2 

261.0 

400.2 

127.8 

262.0 

400.8 

128.0 

262.4 

401.0 

128.3 

263  .  0 

401  .  3 

128.9 

264.0 

401.9 

129.0 

264.2 

402.0 

129.4 

265  .  0 

402.4 

130.0 

266.0 

403.0 

130.6 

267.0 

403.6 

131.0 

267.8 

404.0 

131°  C.  amyl  alcohol  boils 

131.1 

268.0 

404  .  1 

131.7 

269.0 

404.7 

132.0 

269.6 

405.0 

132.2 

270.0 

405.2 

132.8 

271.0 

405.8 

133.0 

271.4 

406.0 

133.3 

272.0 

406.3 

133.9 

273.0 

406.9 

134.0 

273.2 

407.0 

134°  C.  A1(NO3)3+  9H20  boils 

134.4 

274.0 

407.4 

135.0 

275.0 

408.0 

135.6 

276.0 

408  .  6 

136.0 

276.8 

409.0 

136.1 

277.0 

409.1 

136.4°  C.  alloy  Sn3Bi4  melts 

136.7 

278.0 

409.7 

137.0 

278  .  6 

410.0 

137.2 

279.0 

410.2 

137.8 

280.0 

410.8 

138.0 
138.3 

280.4 

281  .0 

411.0 
411.3 

138.12°  C.  sulphur  monochloride,  at  760  mm  pres 
[sure,  boila 

138.9 

2S2  .  0 

411.9 

139.0 

282.2 

412.0 

139.4 

283.0 

412.4 

140.0 

284.0 

413.0 

40°  C.  ferrous  sulphate  -f-  6  aq.  melts 

140.6 

285  .  0 

413.6 

141.0 

285  .  8 

414.0 

141.1 

286.0 

414.1 

141.7 

287.0 

414.7 

142.0 

287.6 

415.0 

142.2 

288.0 

415.2 

142.8 

289.0 

415.8 

143.0 

289.4 

416.0 

143.3 

290.0 

416.3 

160 


THERMOMETER    SCALES. 


THERMOMETER  SCALES— (Continued). 


Centi- 
grade 
Deg. 

Fahren 
heit 
Deg. 

Abso- 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate). 

143.9 

291.0 

416.9 

144.0 

291.2 

417.0 

144.4 

292.0 

417.4 

145.0 

2*93.0 

418.0 

145.6 

294.0 

418.6 

146.0 

294.8 

419.0 

146°  C.  SnL,  melts 

146.1 

295.0 

419.1 

146.3°  C.  alloy  CdBi4  melte 

146.7 

296.0 

419.7 

147.0 

296.6 

420.0 

147.2 

297.0 

420.2 

147.8 

298.0 

420.8 

148.0 

298.4 

421.0 

148.3 

299.0 

421.3 

148.9 

300.0 

421.9 

149.0 

300.2 

422.0 

149.4 

301.0 

422.4 

• 

150.0 

302.0 

423.0 

152.0 

305  .  6 

425.0 

154.0 

309.2 

427.0 

156.0 

312.8 

429.0 

158.0 

316.4 

431.0 

160.0 
162.0 

320.0 
323.6 

433.0 
435.0 

160°  C.  pure  sugar  melts 
161°  C.  turpentine  boils 

164.0 

327.2 

437.0 

166.0 

330.8 

439.0 

168.0 

344.4 

441.0 

170.0 
172.0 

338  .  0 
341.6 

443.0 
445.0 

170°  C.  copper  nitrate,  (Cn(NO,  V*H2O),  bo?J 

174.0 

345  .  2 

447.0 

173.8°  C.  alloy  CdSn2  melts 

176  0 

348.8 

449.0 

175°  C.  ordinary  camphor,  (C10H16O),  melts 

178.0 

352.4 

451.0 

[78°  C.  caffeine  melts 

180.0 

356.0 

453.  G 

80°  C.  aluminium  chloride  boils 

182.0 

359.6 

455.0 

80°  C.  lithium  melts 

184.0 

363  .  2 

457.0 

81°  C.  aniline  boils 

186.0 

366.8 

459.0 

181°  C.  alloy  PbSn.  melts 

188.0 

370.4 

461.0 

182°  C.  phenol  (carbolic  acid)  boils 

190.0 

374.0 

463.0 

187°  C.  alloy  PbSn4  melts 

192.0 

377.6 

465.0 

194.0 

381.2 

467.0 

196.0 

384.8 

469.0 

198.0 

3SS  .  4 

471.0 

97°  C.  alloy  PbSn2  melts 

200.0 
205.0 
210.0 

392.0 
401.0 
410.0 

473.0 
478.0 
483.0 

204°  C.  ordinary  camphor,  (Ci0Hi6O),  boils 
16.4°-216.8°  C.  naphthalene  boils 
17°  C.  crystalline  selenium  insol.  in  CS-,  meltfl 

215.0 

419.0 

488.0 

18°  C.  silver  nitrate  melta 

220.0 
225.0 

428.0 
437.0 

493.0 
498.0 

21°  C.  very  faint  yellow  in  tempering  steel 
30°  C.  silver  chlorate  melts 

230.0 
235.0 

446.0 
455.0 

503.0 
508.0 

32°  C.  pale  straw  yellow  in  tempering  pteel 
35°  C.  tin  melts 

240.0 

464.0 

513.0 

35°  C.  alloy  PbSn  melts 

THERMOMETER   SCALES. 


161 


THERMOMETER  SCALES— (Continued). 


Centi- 
grade 
Deg. 

Fahren- 
heit 
Deg. 

Abso- 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate). 

215.0 

473.0 

518.0 

243°  C.  full  yellow  color  in  tempering  steel 

250.0 

482  .  0 

523.0 

251°  C.  silver  chloride  melts 

254.0 

489  .  2 

527.0 

254°  C.  brown  color  in  tempering  steel 

255.0 

491.0 

528.0 

255°  C.  red  phosphorus  melts 

260.0 

500.0 

533.0 

262°  C.  zinc  chloride  melts 

265.0 

509.0 

538.0 

266°  C.  red  color  in  tempering  steel 

270.0 

518.0 

543.0 

269°  C.  bismuth  melts 

275.0 

527.0 

548.0 

270°  C.  alloy  Pb2Sn  melts 

280.0 

536.0 

553  .  0 

277°  C.  purple  color  in  tempering  steel 

283.0 

541.4 

556.0 

283°  C.  alloy  Pb3Sn  melts 

285.0 

545.0 

558.0 

275°-280°  C.  glycerine  distills 

288.0 

550  .  4 

561.0 

287.3°  C.  yellow  phosphorus  boils  at  762  mm 

290.0 

554.0 

563.0 

288°  C.  mercuric  chloride  melts 

295.0 

563.0 

568.0 

288°  C.  bright-blue  color  in  tempering  steel 

300.0 

572.0 

573.0 

292°  C.  alloy  Pb4Sn  melts 

305.0 
310.0 

581.0 
590.0 

578.0 
583.0 

293°  C.  full  blue  color  in  tempering  steej 
302°  C.  sodium  chlorate  melts 

315.0 

599.0 

588.0 

303°  C.  mercuric  chloride  boils 

320  .  0 

608.0 

593.0 

316°  C.  sodium  nitrate  melts 

325.0 

617.0 

598.0 

316°  C.  dark-blue  color  in  tempering  steel 

330.0 

626.0 

603.0 

320°-327°  C.  cadmium  melts 

335.0 

635  0 

608.0 

327°  C.  lead  melts 

340.0 

644  0 

613.0 

338°  C.  pure  sulphuric  acid,  sp.  gr.  1.854,  boils 

345.0 

653.0 

618.0 

339°  C.  potassium  nitrate  melts 

350.0 

662.0 

623.0 

357.25°  C.  mercury  boils 

360.0 

680.0 

633.0 

359°  C.  potassium  chlorate  melts 

370.0 

698.0 

643.0 

380.0 

716.0 

653.0 

390.0 

734.0 

663  .  0 

400.0 

752.0 

673.0 

410.0 

770.0 

683  .  0 

414°  C.  barium  chlorate  melts 

420.0 

788.0 

693  .  0 

419°  C.  zinc  melts  (Berthelot) 

430.0 

806  .  0 

703  .  0 

434°  C.  cuprous  chloride  melts 

440.0 

824.0 

713.0 

445°  C.  sulphur  boils  (Berthelot) 

450.  u 

842.0 

723.0 

446°  C.  tellurium  melts 

460.0 

860.0 

733.0 

448.4°  C.  sulphur  at  760  mm  boils  (Carnelley) 

470.0 

878.0 

743.0 

480.0 

896.0 

753  .  0 

482°  C.  sodium  perchlorate  melts 

490.0 

914.0 

763.0 

486°  C.  silver  perchlorate  melts 

500.0 

932.0 

773.0 

498°  C.  lead  chloride  and  cupric  chloride  melt 

510.0 
520  .  0 

950.0 
968.0 

783.0 
793.0 

505°  0.  barium  perchlorate  melts 
525°  C.   first   visible   red   of  incandescent    bodies 

530.0 

986.0 

803.0 

[(Pouillet) 

540.0 

1004.0 

813.0 

541°  C.  cadmium  chloride  melts 

550.0 

1022.0 

823.0 

560.0 

1040.0 

833.0 

561°  C.  calcium  nitrate  melts 

570.0 

1  058  .  0 

843.0 

561°  C.  borax  melts 

580.0 

1076.0 

853.0 

590.0 

1094.0 

863.0 

600.0 

1  112.0 

873.0 

598°  C.  lithium  chloride  melts 

162 


THERMOMETER   SCALES. 


THERMOMETER  SCALES— (Continued). 


Centi- 
grade 
Deg. 

Fahr 
Deg. 

Abso 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate). 

610 

1  130 

883 

610°  C.  potassium  perchlorate  melts 

620 

1  148 

893 

617°-6280  C.  stannous  chloride  boils 

630 
640 

1  166 
1  184 

903 
913 

632°  C.  antimony  melts 

650 

1  202 

923 

650°  C.  NaCl  andKCl  in  molecular  proportions  melt 

660 

1  220 

933 

657°  C.  aluminium  melts 

670 
680 

1  238 
1  256 

943 
953 

664°-666°  C.  crystalline  selenium  boils  at  760  mm 
676°-683°  C.  zinc  chloride  boils 

690 

1  274 

963 

700 

1  292 

973 

700°  C.  (about)  dull-red  incandescence  (Pouillet) 

710 

1  310 

983 

708°  C.  magnesium  chloride  melts 

720 
730 
740 

1  328 
1  346 
1  364 

993 
1  003 
1013 

719°  C.  calcium   chloride,  (CaCl2),  melts  (see   also 
719°-731°  C.  potassium  boils                             [28.5°  C.) 
734°  C.  potassium  chloride  melts 

750 

1  382 

1  023 

750°  C.  magnesium  melts 

760 

1  400 

1  033 

763°-772°  C.  cadmium  boils 

770 

1  418 

1043 

772°  C.  sodium  chloride  (kitchen  salt)  melts 

780 

1  436 

1  053 

790 

1  454 

1063 

800 

1472 

1073 

800°  C.  incipient  cherry-red  (Pouillet) 

810 

1490 

1083 

*14°  C.  sodium  carbonate  melts 

820 

1508 

1  093 

U8°  C.  lithium  sulphate  melts 

830 

1  526 

1  103 

834°  C.  potassium  carbonate  melts 

840 

1  544 

1  113 

847°  C.  alloy  63%  silver  +  37%  copper  melts 

850 

1562 

1  123 

850°  C.  alloy  75%  silver  +  25%  copper  melts 

860 

1  580 

133 

$61°  C.  sodium  sulphate  melts 

8.70 

1598 

143 

870.5°  C.  alloy  71.9%  silver  +  28.1%  copper  melts 

880 

1  616 

153 

886°     C.     "     82.1%      "     +17.9%       " 

890 

1  634 

163 

900°  C.        "     57%         "     +43% 

900 

1  652 

173 

900°  C.  cherry-red  incandescence  (Pouillet) 

920 

1688 

193 

'02°  C.  calcium  fluoride  melts 

940 

1724 

213 

950°  C.  zinc  boils 

960 

1760 

233 

968°  C.  silver  melts 

980 
1  000 

1796 
1832 

253 
273 

975°  C.  alloy  60%  silver  +  40%  gold  melta 
000°  C.  bright  cherry-red  (Pouillet) 

I  020 

1868 

1293 

015°  C.  potassium  sulphate  melts 

1040 

1  904 

1  313 

050°-1  100°  C.  white  pig  iron  melts 

1  060 

1  940 

1  333 

064°  C.  gold  melts  (Berthelot) 

1  080 

1976 

1  353 

084°  C.  copper  melts 

1  100 

2012 

1373 

100°  C.  dull-orange  incandescence  (Pouillet) 

1  120 

2048 

1  393 

100°  C.  alloy  95%  gold  +5%  platinum  melts 

I  140 

2084 

1413 

130°  C.     "     90%    "    +10% 

1  160 

2  120 

1  433 

150°  C.  limit  of  gas  thermometers,  bulb  softens 

1  180 
1  200 

2  156 
2  192 

1  453 
1473 

190°  C.  alloy  80%  gold  +  20%  platinum  melts 
200°  C.  bright-orange  incandescence  (Pouillet) 

1220 

2228 

1  493 

200°  C.  (about)  gray  pig  iron  melts 

1  240 

2264 

1  513 

245°  C.  manganese  99%  pure  melts  (Heraeus) 

1  260 

2300 

1  533 

255°  C.  alloy  70%  gold  +  30%  platinum  melts 

1  280 

2336 

1  553 

285°  C.     "     65%     "    +35% 

1  300 

2372 

1573 

300°  C.  white  incandescence  (Pouillet) 

THERMOMETER    SCALES. 


THERMOMETER  SCALES— (Concluded). 


163 


Centi- 
grade 
Deg. 

Fahr. 
Deg. 

Abso- 
lute 
Deg. 

Concrete  Scale  (mostly  only  approximate). 

1  320 

2408 

1  593 

1  320°  C.  alloy  60%  gold  +  40%  platinum  melts 

1340 

2444 

1613 

1350°C.     "     55%     "    +45% 

1360 

2480 

1  633 

1  350°  C.  Bunsen  flame;  inner  blue  flame  (Kelvin) 

1  380 

2516 

1653 

1  370°  C.  furnace  temperature  for  hard  porcelain 

1400 

2  552 

1673 

1  375°  C.  glass  furnace  temperature 

1  420 

2588 

1  693 

1  400°  C.  bright-white  in  andescence  (Pouillet) 

1440 

2624 

1713 

1  400°  C.  Bessemer  steel  melts 

1  460 

2660 

1733 

1  460°  C.  alloy  40%  gold  +  60%  platinum  melts 

1480 

2696 

1753 

1  490°  C.  Siemens-Martin  steel  melts 

1  500 

2732 

1773 

1  500°  C.  dazzling-  white  heat  (Pouillet) 

1520 

2768 

1793 

1  500°  C.  palladium  melts;  tin  boils 

1  540 

2804 

1813 

1  535°  C.  alloy  30%  gold  +70%  platinum  melts 

1  560 
1  580 

2840 
2876 

1833 
1853 

1  425°-l  625°  C.  temp.  Argand  gas-burner  (Lum  ner) 
1  490°-1  580°  C.  temp,  in  Siemens-Martin  furnace 

1  600 

2912 

1873 

1  475°-l  685°  C.  temp,  candle-flame  (Lummer) 

1  620 

2948 

1893 

1  610°  C.  alloy  20%  gold  +80%  platinum  melts 

1  640 

2984 

1  913 

1650°C.     "     15%     "    +85% 

1660 
1  680 

3020 
3056 

1  933 
1  953- 

1  600°-1  825°  C.  temp.  inc.  eleo.  lamp  (Lummer) 
1  690°  C.  alloy  10%  gold  +  90%  platinu.n  melts 

1700 

3092 

1  973 

1  700°  C.  bismuth  boils 

1720 

3  128 

1  993 

1  730°  C.  alloy  5%  gold  +  95%  platinum  melts 

1740 

3  164 

2013 

1  600°-1  800°  C.  thallium  boils 

1760 

3200 

2033 

1780 

3236 

2053 

1  779°  C.  platinum  melts 

1800 

3272 

2073 

1  800°  C.  cobalt  melts 

1825 

3317 

2098 

1  820°  C.  Bunsen  flame  blast-lamp  (Kelvin) 

1  850 

3  362 

2  123 

1875 

3407 

2  148 

1  800°-2  000°  C.  pure  wrought  iron  melts 

1  900 

3452 

2173 

1  900°  C.  manganese  melts 

1  925 

3497 

2  198 

1  950 

3542 

2223 

1  950°  C.  iridium  melts 

1  975 

3  587 

2  248 

2000 

3  632 

2273 

1  800°-2  000°  C.  pure  wrought  iron  melts 

2  100 

3  812 

2  373 

2  000°  C.  rhodium  melts 

2  500 

4532 

2773 

1  925°-2  175°  C.  temp,  of  Nernst  lamp  and  Welsbach 

[mantel  (Lummer) 

3000 

5  432 

3273 

2  500°  C.  osmium  melts 

3  500 

6  332 

3773 

2  844°  C.  oxyhydrogen  flame  (Bunsen) 

4000 

7  232 

4273 

3  475°-3  927°  C.   temp,  electric   (carbon)  arc  lirrht 

5  000 

9  032 

5  273 

f(Lummer) 

6000 

0832 

6273 

6  000°  C.  temperature  of  the  sun  (Lummer) 

Exceedingly  high  temperatures  of  the  order  of  millions  of  degrees  have 
been  calculated  to  be  produced,  on  certain  theoretical  assumptions,  by  the 
impact  of  some  kinds  of  rays,  like  the  X-rays;  but  owing  to  the  exceed- 
ingly small  quantity  of  the  heat  and  the  enormously  rapid  conduction  they 
cannot  be  maintained. 

Note. — Any  inconsistencies  that  may  appear  in  some  of  the  high-temper- 
ature values  in  the  last  column  are  no  doubt  due  in  part  to  the  different 
methods  by  which  they  were  determined;  the  values  are  generally  only 
approximations. 


164  MONEY. 


MONEY. 

The  values  of  foreign  moneys  in  the  terms  of  the  U.  S.  gold  dollar,  as 
given  below,  are  those  used  by  the  United  States  Treasury  Department  in 
1903  for  estimating  the  value  of  all  foreign  merchandise  exported  to  the 
United  States  expressed  in  the  metallic  currencies  of  those  countries.  (See 
Department  Circular  No.  37  of  1903  of  the  Treasury  Department.)  The 
values  refer  to  the  standard  specie  money  of  those  countries  and  were 
determined  by  the  U.  S.  Mint  in  terms  of  the  U.  S.  gold  dollar;  they  are 
therefore  the  true  relative  values,  gold  being  the  basis  of  comparison.  The 
reciprocal  values  were  calculated  from  these.  The  coins  of  silver-standard 
countries  are  valued  by  their  pure  silver  contents  at  the  average  market  price 
of  silver  during  the  beginning  of  the  year  1903.  Much  of  the  money  in 
foreign  countries  is  paper  money,  the  value  of  which  is  subject  to  many 
fluctuations,  and  is  generally  not  received  at  its  par  value. 

Unless  otherwise  stated  the  standard  of  the  countries  in  the  following 
list  is  gold.  France,  Belgium,  Italy,  Switzerland,  and  Greece  form  the 
Latin  Union;  their  standard  coins  have  the  same  value. 

Argentine  Republic.  1  peso  =  0.965  U.  S.gold  dollar- 100  r-entavos. 
1  argentine  (gold)  =  5  pesos  =  4.824  U.  S.gold  dollars.  1  U.  S.  gold  dollar=- 
1.036  pesos. 

Austria-Hungary.  1  crown  (Krone)  =  0.203  U.  S.  gold  dollar.  1 
U.  S.  gold  dollar  =  4. 92;»  rrowns.  The  small  unit  is  tne  heller.  The  former 
units  were  1  florin  =  0.482  U.  S.  gold  dollar  =  100  kreutzer.  (The  usual 
exchange  value  is  much  lower,  being  about  41  cts.,  and  fluctuates  consider- 
ably.) Money  orders  issued  in  the  United  States  are  drawn  on  French 
money  (francs);  1  crown  =  about  1.05  francs. 

Belgium.  1  franc  =  0.193  U.S.  gold  dollar  =  100  centimes.  1  U.  S. 
gold  dollar  =  5. 18  francs 

Bolivia  (silver  standard).  1  boliviano  =  0.352  U.  S.  gold  dollar=* 
100  centavos.  1  U.  S.  gold  dollar  =  2. 84  bolivianos. 

Brazil.  1  milreis  =  0.546  U.  S.  gold  dollar=1000  reis.  1  U.  S.  gold 
dollar  =  1.83  milreis. 

British  Honduras.      1  dollar  =  1  U.  S.  gold  dollar  =100  cents. 

British  Possessions  in  North  America  (except  Newfoundland). 
1  dollar  =  1  U.  S.  gold  dollar. 

Canada.     See  British  Possessions. 

Central  American  States.     See  individual  States. 

Ceylon.      Rupee.     See  India. 

Chile.  1  peso  =  0.365  U.  S.  gold  dollar  =  10  dineros  or  decimos  =  100 
centavos.  1  U.  S.  gold  dollar  =  2.740  pesos.  1  condor  (gold)  =7.300  U.  S. 
gold  dollars  =  2  doubloons  =  4  escudos  =  20  pesos. 

China  (silver  standard).  1  tael  (average  value)  =  0.549  U.  S  gold  dollar. 
1  U.  S.  gold  dollar -=1.82  taels  (average  value).  The  tael  has  different 
values  in  different  cities,  varying  from  0.520  to  0.580  dollar.  The  "British 
dollar"  has  the  same  legal  value  as  the  Mexican  dollar  in  Hong-kong,  the 
Straits  Settlements,  and  Labuan. 

Colombia  (silver  standard).  1  peso  =  0.352  U.  S.  gold  dollar  =  10 
diernos  or  rfecimos  =*  100  centavos.  1  U  S  gold  dollar  =  2.84  pesos.  1 
condor  (gold)  =  9.647  U.  S.  gold  dollars. 

Costa  Iliosi.  1  colon  =  0.465  U.  S.  gold  dollar  =100  centimes.  1  U.  S. 
gold  dollar  =  2.15  colons. 

Cuba.  1  peso  =  0.926  U.  S.  gold  dollar.  1  U.  S.  gold  dollar  =  1.080 
pesos.  1  doubloon  Isabella,  cen ten  =  5.017  U  S.  gold  dollars.  1  Alphonse  = 
4.823  U.  S.  gold  dollars. 

Hi-nmark.  1  crown  (Krc  ne)  =  0.268  U.  S.  gold  dollar  =100  oere.  1  U.  S. 
gold  dollar  =  3.73  crowns. 

Ecuador.     1  sucre  =  0.487  U.  S.  gold  dollar.     1  U.  S-  gold  dollar  =  205 

Egypt.  1  pound  =  4.943  U.  S.  gold  dollars  =  100  piasters  =  4  000  paras. 
1  U.  S.  gold  dollar  =  0.202  pound. 

Finland.  1  mark  (markka)  =  0.193  U.  S.  gold  dollar  =  100  penm. 
1  U.  S  gold  dollar  =  5. 18  marks. 


MONEY.  165 

France.  1  franc  =  0.193  U.  S.  gold  dollar  =  100  centimes.  1  Louis  or 
Napoleon  =  20  francs  1  sou  =  5  centimes  =  about  1  cent  (US.).  1  U  S 
gold  dollar  =  5. 18  francs.  1  U.  S.  cent  =  5  18  centimes  =  about  1  sou  20 
Francs  =  3.86  U  S  gold  dollars.  1  franc  =  0.811  mark  (German)  =  0.7 93 
shilling  (English). 

German  Kfcnpire.  1  mark  =  0.238  U.  S  gold  dollar=100  pfennig  1 
U  S.  gold  dollar  =  4. 20  marks.  1  pfennig  =  0.238  cent;  1  cent  =4.20  pfen- 
nigs. 1  mark  =  1 .233  francs  (French)  =  J.978  shilling  or  1 1  72  pence  (English). 

Great  Britain.  1  pound  sterling  [£]  or  sovereign  =  4. 8665  U.  S.  gold 
dollars  =  20  shillings  =  240  pence  =  960  farthings.  1  U.  S.  dollar  =  0.205  49 
pound  sterling.  1  guinea  =  21  shillings  =  5. 11  U.  S.  gold  dollars.  "1  half 
crown  =  2  5  shillings  =  .608  3  U.  S  gold  dollar.  ]  shilling  [s]  =  24.33  cents  = 
0.05  £  =  12  pence  =  48  farthings  =  1.022  marks  (German)  =  1.261  francs 
(French).  1  U.  S.  gold  dollar  =  4.109  8  shillings.  1  penny  (plural  pence) 
[d]  =  2.()3  cents  -0.0833  shilling  =  0.004167 <£  =  10.51  centimes  (French)  = 
8-52  pfennig  (German).  1  cent  =  0.493  penny. 

Greece.  1  drachma  =  0.193  U.  S.  gold  dollar  =  100  lepta.  1  U.  S.  gold 
dollar  =  5. 18  drachmas. 

Guatemala  (silver  standard).  1  peso  =  0.352  U.  S.  gold  dollar.  1  U.  S. 
gold  dollar  =  2. 84  pesos. 

Haiti.  1  gourde  =0.965  U.  S.  gold  dollar  =100  cents  (Haiti).  1  U.  S. 
gold  dollar  =1.036  gourdes. 

Hawaii.     See  United  States. 

Honduras  (silver  standard).  1  peso  =  0.352  U.  S.  gold  dollar.  1  U.  S. 
gold  dollar  =  2. 84  pesos. 

Honduras  (British).     1  dollar=l  U.  S.  gold  dollar=100  cents. 

India.  1  pound  sterling  [£]=4.866  5  U.  S.  gold  dollars  (same  standard 
as  in  Great  Britain)  =  15  rupees  (money  of  account).  1  rupee  =  0.324  4  U.  S. 
gold  dollar  =  16  annas.  1  U.  S.  gold  dollar  =  0.205  49  pound  sterling  =  3.082 
rupees. 

Italy.  1  lira  =  0.193  U.  S.  gold  dollar  =100  centesimi.  1  U.  S.  gold 
dollar  =  5. 18  lire. 

Japan.  1  yen  =  0.498  U.  S.  gold  dollar=100  sen.  1  U.  S.  gold  dollar  = 
2.008  yen. 

Liberia.     Same  as  in  the  United  States. 

Mexico  (silver  standard).  1  dollar  (silver),  peso,  or  piastre  =  0.383  U.  S 
gold  dollar  =  100  centavos.  1  dollar  (gold)  =  0.983  U.  S.  gold  dollar.  1  U.  S. 
gold  dollar  =  2.611  dollars  (silver)  •=  1 .017  dollars  (gold).  1  once  or  doubloon 
=  16  pesos. 

Netherlands.  1  florin  of  100  cents  =  0.402  U.  S.  gold  dollar.  1  U.  S. 
gold  dollar  =  2. 49  florins. 

Newfoundland.  1  dollar=  1.014  TJ.  S.  gold  dollars.  1  U.  S.  gold  dollar 
=  0,986  dollar. 

Nicaragua  r  silver  standard).  1  peso  =  0.352  U.  S.  gold  dollar.  1  U.  S. 
gold  dollar  =  2. 84  pesos. 

Norway.  1  crown  =  0. 268  U.  S.  gold  dollar  =  100  o~re  =  30  skillings.  1 
U.  S.  gold  dollar  =  3.73  crowns. 

Persia  (silver  standard).  1  kran  (silver) -0.065  U.  S.  gold  dollar.  1 
tomans  (gold)  =  1 .704  5  U.  S.  gold  dollars.  1  U.  S  gold  dollar  =  15.4  krans 
=  0.5867  tomans  (gold). 

Peru.  1  sol -0.487  U.  S.  gold  dollar=10  dineros=100  centavos.  1 
libra  (gold)  =  4.866  5  U.  S.  gold  dollars.  1  U.  S.  gold  dollar  =  2.053  sols  = 
0.205  49  libra  (gold). 

Portugal.  1  milreis  =  1 .080  U.  S.  gold  dollars  =  10  testoons  — 1  000  reis. 
1  crown  =  10  milreis.  1  U.  S.  gold  dollar  =  0.925  9  milreis. 

Russia.  1  ruble  =  0. 515  U.  S  gold  dollar  =  2  poltinniks  =  4  tchetvertaks 
=  5  abassis=10  griviniks  =  20  pietaks=100  kopecks.  1  imperial  (gold)  = 
15  rubles  =7.7 18  U.  S.  gold  dollars.  1  U.  S.  gold  dollar  =1.942  rubles. 
1  kopeck  =  0.51 5  cent  (U.  S.);  1  cent  =  1.942  kopecks. 

Sandwich  Islands.     S^e  United  States. 

Salvador  (silver  standard).  1  peso  =  0.352  U.  S.  gold  dollar.  1  U.  S. 
gold  dollar  =  2. 84  pesos. 

Sicily.      See  Italy. 

Spain.  1  peseta  or  pistareen  =  0.193  U.  S.  gold  dollar=100  centimes. 
1  U.  S.  gold  dollar  =  5. 18  pesetas. 

Sweden.  1  crown  =  0.268 U.S. gold  dollar  =100  oere.  1  U.  S.  gold  dol- 
lar =3. 7  3  crowns. 


166 


MONEY   AND    LENGTH. 


Switzerland.  1  franc  =  0.193  U.  S.  gold  dollar  =  100  centimes.  1  U.  S. 
gold  dollar  =  5. 18  francs. 

Turkey.  1  piaster  =  0.044  U.  S.  gold  dollar  =  40  paras.  1  U.  S.  gold 
dollar  =  22.73  piasters. 

United  States  (and  possessions).  1  dollar  fS]=  1  U  S.  gold  dollar- 10 
dimes  =100  cents  1  eagle  =  20  dollars. 

Uruguay.  1  peso  =  1.034  U.  S.  gold  dollars  =100  centavos  or  centesi- 
mos.  1  U.  S.  gold  dollar  =  0.967  peso. 

Venezuela.  1  bolivar  =  0.193  U.  S  gold  dollar  =  2  decimos.  1  U.  S 
gold  dollar  =-5. 18  bolivars.  1  venezolano  =  5  bolivars.  . 


FLUCTUATING  CURRENCIES. 

The  November,  1903,  issue,  No.  278,  vol.  73,  of  the  Monthly  Consular 
Reports,  published  by  the  Department  of  Commerce  and  Labor,  gives  a 
list  of  the  fluctuating  values  of  the  silver  units  of  several  countries.  The 
latest  values,  namely  those  for  October  1,  1903,  are  as  follows: 

Bolivia,  silver  boliviano  =  $0.408. 

Central  America,  silver  peso  =  $0.408. 

China,  tael,  average  value  =  $0.636. 

Colombia,  silver  peso  =  $0.408. 

Mexico,  silver  dollar  =  $0.443. 

Persia,  silver  kran  =  $0.075. 


MONEY  and  LENGTH. 

The  money  values  used  in  this  table  are  those  given  in  the  above  Hit. 
1  shilling  per  mile=     0.784  franc  per  kilometer. 
=     0.635  mark  per  kilometer. 
=  0.243  3  dollar  per  mile. 
1  franc  per  kilometer  =  1.276  shillings  per  mile. 

=  0.811  mark  per  kilometer. 
=  0.311  dollar  per  mile. 
1  mark  per  kilometer  =  1.575  shillings  per  mile. 

=  1.233  francs  per  kilometer. 
=  0.383  dollar  per  mile. 
I  dollar  per  mile  =  4. 11  shillings  per  mile. 

=  3.22  francs  per  kilometer. 
=  2.61  marks  per  kilometer. 
1  ~«mt  per  foot  =  0.493  penny  per  foot. 
=  0.170  franc  per  meter. 
=  0.138  mark  per  meter. 
1  penny  per  foot  =   2.03  cents  per  foot. 
=0.345  franc  per  meter. 
=  0.280  mark  per  meter. 
1  franc  per  meter  =   5  88  cents  per  foot. 
=   2.90  pence  per  foot. 
=  0.811  mark  per  meter. 
1  mark  per  meter  =  7.26  cents  per  foot. 
=  3.57  pence  per  foot. 
=  1.233  francs  per  meter. 


MONEY    AND  WEIGHT.  167 


MONEY   and   WEIGHT. 

The  money  values  used  in  this  table  are  those  given  in  the  above  list. 
Av.  means  avoirdupois  weights. 

1  franc  per  metric  ton  =  0  8H  mark  per  metric  ton. 
=  0.806  shilling  per  long  ton. 
=  0.720  shilling  per  short  ton. 
=  0.196  dollar  per  long  ton. 
=  0.175  dollar  per  short  ton. 

1  mark  per  metric  ton  =  1.233  francs  per  metric  ton. 
=  0.994  shilling  per  long  ton. 
=  0.888  shilling  per  short  ton. 
=  0.242  dollar  per  long  ton. 
=  0.216  dollar  per  short  ton. 

1  shilling  per  long  ton  =    1.24  francs  per  metric  ton. 
=  1.005  marks  per  metric  ton. 
=  0.243  dollar  per  long  ton. 
=  0.217  dollar  per  short  ton. 

1  shitling.  per  short  ton=    1.39  francs  per  metric  ton. 
=    1.13  marks  per  metric  ton. 
=  0.273  dollar  per  long  ton. 
=  0.243  dollar  per  short  ton. 

1  dollar  per  long  ton  =5.10  francs  per  metric  ton. 
=  4.14  marks  per  metric  ton. 
=  4.11  shillings  per  long  ton. 
=  3.67  shillings  per  short  ton. 

1  dollar  per  short  ton  =  5.70  francs  per  metric  ton. 
=  4.63  marks  per  metric  ton. 
=  4.60  shillings  per  long  ton. 
=  4.11  shillings  per  short  ton. 

1  franc  per  kilogram  =     0.811  mark  per  kilogram. 
=    0.547  cent  per  ounce  (av.). 
=     0.360  shilling  per  pound  (av.). 
=     0.270  penny  per  ounce  (av.). 
=  0.087  5  dollar  per  ounce  (av.). 
1  mark  per  kilogram  =  1.233  francs  per  kilogram. 
=  0.675  cent  per  ounce  (av.). 
=  0.444  shilling  per  pound  (av.). 
=  0.333  penny  per  ounce  (av.). 
=  0.108  dollar  per  pound  (av.).     . 
1  cent  per  ounce  [av.]=    1.83  francs  per  kilogram. 
=    1.48  marks  per  kilogram. 
=  0.658  shilling  per  pound  (av.). 
=  0.493  penny  per  ounce  (av.). 
=  0.160  dollar  per  pound  (av.). 

1  shilling  per  pound  [av.]=    2.78  francs  per  kilogram. 
=   2.25  marks  per  kilogram. 
=    1.52  cents  per  ounce  (av.). 
=  0.750  penny  per  ounce  (av.). 
=  0.243  dollar  per  pound  (av.). 

1  penny  per  ounce  [av.]=   3.71  francs  per  kilogram. 
=   3.00  marks  per  kilogram. 
=   2.03  cents  per  ounce  (av.). 
=  1.333  shillings  per  pound  (av.). 
=  0.324  dollar  per  pound  (av.). 

I  dollar  per  pound  [av.]=  11.4  francs  per  kilogram. 
=  9.26  marks  per  kilogram. 
•=6.25  cents  per  ounce  (av.). 
=  4.11  shillings  per  pound  (av.). 
«=3.08  pence  per  ounce  (av.). 


168 


SCALES   OF   MAPS   AND    DRAWINGS. 


SCALES   of  MAPS   and   DRAWINGS. 

The  following  table  is  limited  to  the  more  usual  values.  For  a  very 
complete  table  of  scales  for  maps,  as  distinguished  from  detail  drawings, 
see  Haupt,  Scales  of  Maps,  Proceedings  Engineers1  Club  of  Philadelphia, 
Vol.  1,  p.  47-59,  1879.  The  relation  of  the  meter  to  the  foot  which  is  there 
used  is  slightly  different  from  the  legal  value  in  this  country;  the  follow- 
ing are  all  based  on  the  standard  legal  value. 

When  the  scale  of  a  map  is  given  as  — .  then : 

n 


1  inch  on  the  map  =n  X 

=nX 
=nX 
=nX 
=nX 
=nX 
=nX 


Logarithm 

2.540005  centimeters 0-4048348 

1 .  inches 0  -000  0000 

0.083  333  3  feet 2  920  8188 

0.027  777  8  yards 2  443  6975 

0.025  400  05  meters 2  404  8348 

0.005  050  51   rods 3.703  3348 

0.001  262  63  chains £.101  2749 


1  centimeter  on  the  map  =n  X 

=nX 
=nX 
=nX 
=nX 
=nX 
=nX 
=nX 


=  n  X 0.000  025  400  05  kilometers 5.404  8348 

=n  X   0.000  015  782  8  miles 5  198  1849 


1.  centimeters  ,  0  000  0000 

0.393  700  inches 1.595  1654 

0.032  808  3  feet 2  515  9842 

0.0109361   yards 20388629 

0.01   meters. 2-000  0000 

0.001  988  38  rods.  .    3. 298  5002 

0.000  497  096  chains 1-696  4403 

0.000  01  kilometers  .  .    5-000  0000 


=nX  0.000  006  213  70  miles 6-7933503 

Special  values   of  the  above,    frequently   used.       The   first    length 
refers  to  the  actual  distance  and  the  second  to  the  map  or  drawing. 


Scale  of  1/12 
1/120 
1/198 
1/240 
1/360 
1/480 
1/600 
1/720 
1/960 
1/120  0 
1/63  360 


1  foot  to  the  inch. 
=  10  feet  to  the  inch. 
=  1  rod  to  the  inch. 
=  20  feet  to  the  inch. 
=  30  feet  to  the  inch. 
=  40  feet  to  the  inch. 
=  50  feet  to  the  inch. 
=  60  feet  to  the  inch. 
=  80  feet  to  the  inch. 
=  100  feet  to  the  inch. 

1  mile  to  the  inch. 


1/100  000=      1  kilometer  to  the  centimeter. 
Scales  for  Detail  Drawings.     The  first  length  refers  to  the  drawing 
and  the  second  to  the  actual  length. 


Full  size 
Half  size 

Third  size  = 

Quarter  size  = 

Sixth  size  = 

Eighth  size  = 

Twelfth  size  => 

Sixteenth  size  = 
Twenty-fouth  size  = 
Thirty-second  size  =• 
Forty-eighth  size  = 
Ninety-sixth  size  =» 


=    12  inches  to  the  foot. 
6       "       "     "       " 
4 
3 
2 


. 

inch 


PAPER  MEASURE.     MISCELLANEOUS  MEASURES. 


1  quire  =  24  or  25  sheets. 

1  ream  =20  quires  =  480  sheets. 

1  ream  =  generally  500  sheets. 

1  bundle  (obs.)  =  2  reams  =  1  000  sheets. 

1  hale  (obsolete)  =  5  bundles. 


1  dozen  =12. 

1  gross  =12  dozen 

1  great  gross  =  12  gross. 


=  144. 


1  jjreat  gross  =  144  dozen  =  1  728. 
1  score  =20. 


FUNCTIONS   OF  It.  169 


USEFUL   FUNCTIONS   OF  7t. 

Logarithm 

Aprx.  means  within  2%. 
•K  (called  "  pi"  )=  circumference  of  a  circle  divided  by  the  diameter  and 

is  a  constant. 

JT  =  approximately  2%,  which  equals  3.142  86  or  ^100%  too  much. 
TT  =  approximately  85^iia,  which  equals  3.141  592  9. 
n=   3.141  592  653  589793  238  462  643  383  279  502  88. t  .  .  .   0  497  1499 

2^  =   6.283  185  307  180 0  798  1799 

3*=   9.424777960769 09742711 

4?r=  12.566  370  614359.    Aprx.  HX 100 1-0992099 

5*=  15.707  963  267  949 1-196  1199 

6;:=  18.849  555  921  539 1-275  3011 

7»  =  21.991  148  575  129 1-3422479 

8*  =  25.132741  228718 1-400  2398 

9^  =  28.274  333  882  308 1-451  3924 

10^  =  31.415  926  535  898 1-497  1499 

471/3  =   4.188  790  204  786 0  622  0886 

jr/1  =3.141  5927.    Aprx.  2% 04971499 

jr/2  =  1.5707963.    Aprx.  !%• 01961199 

7r/3  =  1.047  197  6 0020  0286 

ir/4  =  0.785  398  2.    Aprx.  8/io I  895  0899 

*/5  =  0.628  318  5 1-798  1799 

*/6  =  0.5235988 ' 1-718  9986 

7r/7=0.4487990 1-6520518 

»/8  =  0.392  699  1 1-594  0599 

7r/9  =  0.349  065  9 1-542  9074 

7r/10  =  0.314  159  3    1-497  1499 

jr/12  =  0.261  799  4 1-417  9686 

jr/16  =  0.196  349  5 1-293  0298 

7r/32  =  0.098  174  8 2  991  9999 

ff/64  =  0.049 1)87  4 2  690  9698 

K/108  =  0.0290888 2463  7261 

n-/180  =  0.017  453  3 .  2  241  8774 

7r/360  =  0.008  726  65 3  940  8474 

1  x  7r/4  =  0.785  398  2.    Aprx.8/io I  895  0899 

2X7r/4  =  1.5707963 0  196  1199 

3  X  7T/4  =  2.356  194  5 0  372  2112 

4X7r/4  =  3.141  5927 04971499 

5  x  7T/4  =  3.926  990  8 0  594  0599 

6X;r/4  =  4.7123890 0  673  2411 

7  X  ;r/4  =  5.497  787  1 0  740  1879 

8X;r/4  =  6.283  1853 0  798  1799 

9X7r/4=7.0685835 0  849  3324 

10X7r/4=7.853981  6 0  895  0899 

l/7r  =  0.318  309  9.    Aprx.  %2 I  502  8501 

2/7r  =  0.636  619  8 I  803  8801 

3/;r  =  0.9549297 1  979  9713 

4/7r=  1.273  2395 0  104  9101 

5/;r  =  1.591  5494 0201  8201 

6/^  =  1.9098593 0-281  0014 

7/7r  =  2.228  169  2 0347  9482 

8/7r  =  2.546  479  1 0-405  9401 

9/7r  =  2.864  7890 - 0-4570928 

10/»r  =  3.183  098  9 0-502  8501 

l2/;r  =  3.819  718  6 0-582  0314 


t  Ludolph's  value.     For  Vega's  value  to  140  decimal  places  see  Ilistoire 
des  Recherchea  aur  la  Quadrature  du  Cercle,  by  Montucla,  1831,  p.  282 


170  FUNCTIONS   OF    it. 

Logarithm 

jrv/2     =     4.4428829 0-647.6649 

;T-H  x/2  =     2.221  441  4 • 0-346  6349 

4*-*- 10   =     1.256637061.    Aprx.  add  M 00992099 

n2          =     9.869604401089.    Aprx.  10 0-9942997 

4;r2          =39.478418.    Aprx.  40 1-5963597 

x2+4    =     2.46740110.    Aprx.^XlO 0-3922397 

7T3          =    31.006  276  680  300 1-491  4496 

;r4          =   97.409  091 1-988  5995 

TT>          =306.019  69 2  485  7494 

TTC          =961.389  19 2  982  8992 

1  •*•  x*          =     0.101  321  2 1  005  7003 

1 -J-  n3          =     0.032  251  5 2  508  55U4 

X/*  =      1.772453850908 ,...02485749 

2V*  =      3.544  907  6 0  549  6049 

v/27r  =     2.506  628 : 0  399  0899 

tyn  =      1.464  592 0  165  7166 

1  +   TT  =     0.318  309  866 1-502  8501 

1-^2*  =     0.159154933.    Aprx.  %•*•  10 1-2018201 

l-M;r  =     0.07957747.    Aprx.s/ioo 2-9007901 

10-J-47T  =     0.7957747.    Aprx.s/lo 1-9007901 

10-5-8*  =     0.3978873.    Aprx.^io 15997601 

1^-N/ff  =     0.5641896 1-7514251 

I-J-^TT  =     0.6827841 18342834 

Log   it  =     0.497  149  872  694  133  854  35. 

Log,,  n  =      1.144  729  885  849  400  174  14. 

it  =  180°  considered  as  an  angle. 

7T/180   =     0.017  453  29 2-241  8774 

;r/360   =     0.008  726  65 3  940  8474 

ISO/*  =   57.2957795 1-7581226 

360/7T  =114.591  559  0 2-059  1526 


USEFUL   NUMBERS. 

Logarithm 

2  =  1.414  21.    Aprx.  VrXlO 0-1505150 

=  1.732  05.    Aprx.  % 0-238  5606 

2=  1.259  92.    Aprx.  add  % 0-100  3433 

3  =  1.442  25.    Aprx.  ^X  10 0-1590404 

4  =  1.587  40.    Aprx.%.    0-200  6867 


LOGARITHMS. — PHYSICAL   CONSTANTS.  171 


SYSTEMS   OF  LOGARITHMS. 

The  logarithm  [log]  of  a  given  number  is  the  exponent  which  denotes  the 
power  to  which  a  certain  fixed  numerical  base  is  raised,  in  order  to  produce 
this  given  number.  Thus  if  the  base  is  10,  then  the  log  of  100  is  2,  because 
10  raised  to  the  2d  power  =100.  To  multiply  two  numbers,  add  their 
logs;  to  divide,  subtract  the  log  of  the  divisor  from  that  of  the  dividend; 
then  from  the  resulting  log  find  the  corresponding  number. 

Log  or  logic  means  the  common,  usual,  or  Briggs'  logarithm;  the 
base  of  this  system  is  10. 

Logg  or  In  means  the  Naperian,  natural,  or  hyperbolic  logarithm;  the 
base  of  this  system  is  about  2.718  (see  below),  generally  denoted  by  e. 

Log  of  1  =0  in  any  system. 

Log  of  base  =  l  in  its  own  system. 

e  =  base  of  Naperian,  natural,  or  hyperbolic  logarithms. 

c  =  2.718281  828. 

logio  of  e- 0.434  294  481  903  252. 

Log<>  of  e  =  1 . 

10  =  base  of  usual,  common,  or  Briggs'  logarithms. 

log*  of  10  =  2.302  585  092  994  046. 

Logio  of  10  =  1. 

log*  10Xlog10e  =  l. 

The  modulus  of  any  system  is  the  constant  by  which  the  Naperian 
logarithm  of  a  number  must  be  multiplied  to  give  the  logarithm  of  the 
number  in  that  system.  The  modulus  of  any  system  is  equal  to  the  recip- 
rocal of  the  Naperian  log  of  the  base  of  that  system. 

Modulus  of  Naperian  system  =  1  -v-log*  of  e  =  1. 

Modulus  of  common  system  =  1  •*•  log*  of  10  =  0.434  294  481  903  252. 

The  logarithm  of  a  number  (n)  in  any  system  is  equal  to  the  modulus 
of  that  system  multiplied  by  the  Naperian  logarithm  of  the  number.  Or: 

Logio  n  =  modulus  (common  system)  Xlog^  n. 

Log<j  n  =  modulus  (Naperian  system  =  1)X  log*  n. 

To  change  a  logarithm  from  the  common  to  the  Naperian  system,  or 
the  reverse: 

Common  log  X  2. 302  585  092  994  046  =  Naperian  log. 

Naperian  log  X  0.434  294  481  903  252  =  common  log. 

ACCELERATION   OF   GRAVITY  [g]. 

(See  also  table,  pp.  87,  88.) 

Logarithm 

0  =  9.805  966  meters  per  second  per  second.    Aprx.  10 0-991  4904 

/v/20  =   4.428  54  in  meters  per  second  per  second.    Aprx.  %  X 10.    0  646  2602 

7=   32.1717  feet  per  second  per  second.    Aprx.  32 1-5074746 

=  8.021  44  hi  feet  per  second  per  second.    Aprx.  8 0-904  2523 

MECHANICAL  EQUIVALENT   OF   HEAT   [M]. 

(See  also  p.  72.) 
M—          426.900    kilogram-meters     per     kilogram    calorie. 

Aprx.  %  X 1  000 2  630  3262 

1+M  =  0.002  342  47  in  same  units.    Aprx.%-^1  000 3-369  6738 

M  =         778.104  foot-pounds  per  pound  Fahrenheit  heat-unit. 

Aprx.  %X1  000 2891  0379 

1  -5- M  =  0.001  285  17  in  same  units.    Aprx.  %  -5-1  000 3-1089621 

(For  further  reduction  factors  see  table  of  measures  of  Energy,  pp.  74-77.) 

SPECIFIC   HEAT   OF   WATER. 

(See  also  p.  72.) 
Specific  heat  of  water: 

=  4  186.17  joules  per  kilogram  calorie.    Aprx.  4  200 3-621  8166 

Specific  heat  of  water  (in  joules  per  kg  cal)  =  mechanical 
equivalent  of  heat  (in  kilogram-meters  per  kilogram  calorie) 
X  acceleration  of  gravity  (in  meters  per  second  per  second). 


172  MISCELLANEOUS    FOREIGN    MEASURES. 


MISCELLANEOUS  FOREIGN  MEASURES. 

The  following  list  of  miscellaneous  measures  used  in  foreign  countries' 
was  received  too  late  for  classification  with  the  others.  The  values  are 
taken  from  the  November,  1903,  issue  (No.  278,  vol.  73)  of  the  Monthly 
Consular  Reports  published  by  the  Department  of  Commerce  and  Labor 
of  the  U.  S.  Government.  They  are  the  measures  which  are  referred  to  in 
the  Consular  Reports.  Many  of  them  are  presumably  only  approximately 
correct.  They  have  been  rearranged  here  in  the  alphabetical  order  of  the 
names  of  the  countries.  For  an  alphabetical  list  of  the  names  of  the  meas- 
ures see  the  index. 

Argentine  Republic.  1  pie  =  0.947  8  foot.  1  vara  =  34.1208  inches. 
1  cuadra  =  4.2  acres.  1  frasco  =  2.509  6  quarts.  1  baril  =  20.078  7  gallons. 
1  libra  =«  1.012  7  pounds.  1  arroba  (dry)  =  25.317  5  pounds.  1  quintal  = 
101.42  pounds. 

Belgium.      1  last  =85.134  bushels. 

Bolivia.      1  marc  =  0.507  pound. 

Borneo.      1  picul=  135.64  pounds. 

Brazil.      1  arroba  =  32. 38  pounds.     1  quintal  =  130.06  pounds. 

Celebes.      1  picul  =  135.64  pounds. 

Central  America.  1  vara  =  32.87  inches.  1  centaro  =  4.263  1  gallons. 
1  fanega  (dry)  =  l  574  5  bushels.  1  libra  =  1.043  pounds. 

Chile.  1  vara  =  33.367  inches.  1  fanega  (dry)  =  2.575  bushels.  1  libra 
=  1.014  pounds.  1  quintal  =  101.41  pounds. 

China.  1  tsun  =  1.41  inches.  1  chih=14.  inches.  1  li  =  2115.  feet. 
1  catty  =  1£  pounds.  1  picul  =  133£  pounds. 

Cofhiii  China.      1  tael  =  590.75  grains. 

Costa  Rica.      1  manzana  =  1|  acres. 

Cuba.  1  vara  =  33.384  inches.  1  arroba  (liquid)  =4. 263  gallons.  1 
fanega  (dry)  =  1.599  bushels.  1  libra  =  1.016  1  pounds.  1  arroba  (dry)  = 
25.366  4  pounds. 

Curacao.      1  vara  =  33. 375  inches. 

Denmark.  1  mil  (geographical)  =  4.61  miles.  1  mil  =  4.68  miles.  1 
tondeland  =  1.36  acres.  1  tonde  (cereals)  =  3. 947  83  bushels.  1  centner  = 
110.11  pounds. 

Egypt.  1  pic  =  21i  inches.  lfeddan  =  1.03  acres.  1  ardeb=7.6907 
bushels.  1  oke  =  2.722  5  pounds. 

Greece.  1  drachme=a  half  ounce  (presumably  av.).  1  livre  =  l.l 
pounds.  1  oke  =  2.84  pounds.  1  quintal  =  123. 2  pounds. 

Guiana.      1  livre  =  1.079  1  pounds. 

Holland.      1  last  =85.134  bushels. 

Honduras.      1  milla  =  1.149  3  miles. 

Hungary.      1  oke  =  3.0817  pounds. 

India.  1  bongkal=  832.  grains.  1  seer  =  1  pound  13  ounces.  1  maund 
=  82y  pounds.  1  candy  (Madras)  =  500.  pounds.  1  candy  (Bombay )  = 
529.  pounds. 

Isle  of  Jersey.      1  vergees=71.1  square  rods. 

Japan.  1  bu=0.1  inch.  1  sun  =  1.193  inches.  1  shaku  =  11.930  5 
inches.  1  ken  =  6  feet.  1  tsubo  =  6  feet  square.  1  se=  0.024  51  acres.  1 
tan  =  0.25  acre.  1  sho  =  1.6  quarts.  1  to  =  2  pecks.  1  koku  =  4.9629 
bushels.  1  catty  =  1.31  pounds.  1  picul  =  133$  pounds. 

Java.      1  catty  =  1.35  pounds.     1  picul  =  135.1  pounds. 

Luxemburg.      1  fuder  =  264.17  gallons. 

Malta.  1  caffiso  =  5.4  gallons.  1  barrel  (customs)  =  11. 4  gallons.  1 
cantaro  (cantar)  =  175.  pounds.  1  salm  =  490.  pounds. 

Mexico.  1  vara  =  33.  inches.  1  frasco  =  2.5  quarts.  1  fanega  (dry)  = 
1.54728  bushels.  1  libra=  1.014  65  pounds.  1  quintal  =10 1.41  pounds. 
1  carga  =  300.  pounds. 

Morocco.  1  artel  =  1.12  pounds.  1  fanega  (dry)  strike  =  70  pounds. 
1  cantar  =113.  pounds.  1  fanega  (dry)  full  =  118.  pounds. 

Newfoundland.      1  quintal  (fish)  =  112.  pounds. 

Nicaragua.      1  manzana  =  1|  acres.     1  milla  =  1.149  3  miles. 

Palestine.     1  rottle  =  6  pounds. 


MISCELLANEOUS    FOREIGN    MEASURES.  173 

Paraguay.  1  vara  =  34.  inches.  1  cuadra=78.9  yards.  1  cuadra 
square  =8.077  square  feet  (?).  1  league  (land)  =  4  633.  acres.  1  arobe  =  25. 
pounds.  1  quintal  =  100.  pounds. 

Persia.      1  batman  (tabriz)  =  6.49  pounds. 

Peru.  1  vara  =  33.367  inches.  1  libra  =  1.014  3  pounds.  1  quintal  = 
101.41  pounds. 

Philippine  Islands.      1  picul  =  137.9  pounds. 

Portugal.  1  almuda  =  4.422  gallons.  1  arratel  =  1.0 11  pounds.  1  libra 
=  1.011  pounds.  1  arroba  (dry)  =  32. 38  pounds. 

Russia.  1  arshine  (square)  =  5.44  square  feet.  1  vedro  =  2.707  gallons. 
1  korree  =  3.5  bushels.  1  chetvert  =  5.774  8  bushels.  1  klafter  =  216.  cubic 
feet.  1  funt  =  0.9028  pound.  1  pood  =  36.112  pounds.  1  berkovets  = 
361.12  pounds. 

Russian  Poland.  1  vlocka  =  41.98  acres.  1  garnice=0.88  gallon.  1 
last  =  1  If  bushels. 

Salvador.      1  manzana  =  lg  acres. 

Sarawak.      1  coyan  =  3  098.  pounds, 

Siam.      1  catty  =  1.35  pounds.     1  coyan  =  2667.  pounds. 

Spain.  1  pie  =  0.91407  foot.  1  yara  =  0.914  117  yard.  1  arroba 
(liquid)  =4. 263  gallons.  1  fanega  (liquid)  =  16.  gallons.  1  butt  (wine)  = 
140.  gallons.  1  dessiatine  =  1 .599  bushels.  1  libra  =  1.014  4  pounds.  1 
arroba  (dry)  =  25.36  pounds.  1  frail  (raisins)  =  50.  pounds.  1  barrel 
(raisins)  =  100.  pounds.  1  last  (salt)  =4  760.  pounds. 

Sumatra.      1  l»ouw  =  7  096.5  square  meters.      1  catty  =  2. 12  pounds. 

Sweden.  1  tunna  =  4.5  bushels.  1  pund  =  1.102  pounds.  1  centner  = 
93.7  pounds. 

Syria.  1  rottle  =  5f  pounds.  1  quintal  =  125.  pounds.  1  cantar  (Da- 
mascus) =575.  pounds. 

Turkey.  1  pik  =  27.9  inches.  1  oke  =  2.828  38  pounds.  1  cantar  == 
124.703  6  pounds. 

Uruguay.  1  cuadra  =  nearly  2  acres.  1  suerte  =  2700.  cuadras.  1 
fanega  (single)  =  3. 888  bushels.  1  fanega  (double)  =7. 77 6  bushels.  1  libra 
=  1.014  3  pounds. 

Venezuela.  1  vara  =  33.384  inches.  1  fanega  (dry)  =  1.599  bushels*. 
I  libra  =1.016  1  pounds.  1  arroba  (dry)  =  25.402  4  pounds. 

Zanzibar.     1  frasila="35.  pounds. 


INDEX. 


Ab-,  prefix.  96 

Abassis,  Russia,  165 

Abbreviated  numbers,  9 

Abbreviations,  text,  28 
do.  table  of,  ix 

Abfarad,  117 

Abs-,  prefix,  96 

Absampere,  113 

Abscoulomb,  116 

Absohm,  99 

Absolute: 
potential,  108 
temperature  scale,  151-163 
do.  reduction  factors,  150 
system  of  units,  text,  11 

Absolute  unit  of: 
candle  power,  146 
capacity,  electric,  117 
conductance,  105 
conductivity,  electric,  106,  107 
current,  electric,  112,  113 
electromotive  force,  109,  110 
energy,  electric,  122.  123 
energy,  magnetic,  143 
flux,  magnetic,  138 
do.,  density,  140,  141 
force,  83 
inductance,  119 
light,  3,  145,  146 
magnetic  moment,  142 
magnetization  intensity,  142 
magnetizing  force,  134,  137 
magnetomotive  force,  133 
permeance,  magnetic,  131 
power,  electric,  124,  125 
power,  magnetic,  144 
quantity,  electric.  116 
reluctance,  magnetic,  129 
resistance,  electric,  99 
resistivity,  102,  103 
time,  93 
'Absolute  units: 

Congress  decisions,  14 
dimensional  formulas,  18-26 
generally  called  C.G.S.  units,  11 
list  of,  18-20 
prefixes  for,  96 
relations  to  others,  3 


Absolute  units: 
system  of,  text,  11 
vs.  concrete,  12 

Absolute  values,  elec.  units,  text,  96 
Abstat-,  prefix,  96 
Abstafarad,  117 
Abstatampere,  113 
Abstatcoulomb,  116 
Abstatohm,  99 
Abstat  volt,  110 
Abvolt,  109 
Acceleration: 

angular,  table,  88 

angular,  physical,  19 

gravity,  88,  171 

do.,  mean  value,  87 

do.,  as  a  relation,  3 

linear,  table,  87 

linear,  physical,  19 
Accuracy  of  logarithms,  10 
Accuracy  of  numbers,  9 
Acetate  lamp,  amyl,  144 
Acre,  43 

to  hectares,  digit  table,  44 

circular,  44 

Ireland,  Scotland,  Switzerland,  44 
Acre-foot,  95 
Activity,  19 
Admittance,  table,  105 

defined,  105 

physical,  22,  24 

Almude,  Lisbon,  Oporto,  Constanti- 
nople, 55 

Almuda,  Portugal,  173 
Alnar,  Sweden,  33 
Alphonse,  Cuba,  164 
Alquiere ,  Lisbon ,  Madeira ,  Oporto ,  55 
Alternations,  definition,  86,  121 

number  of,  87,  121 
Am,  Sweden,  55 
Ampere,  table,  113 

definition,  112 

final,  113 

international,  113 

do.,  defined,  112 

Nat.  Bur.  of  Standards,  113 

rate  of  change  of,  per  second,  11? 

Reichsanstalt,  113 

true,  113 

true,  defined,  112 

175 


176 


INDEX. 


Ampere  per  circular  centimeter,  115 
per  circular  inch,  115 

per  circular  mil,  115 

per  circular  millimeter,  115 

per  square  centimeteu,  115 

per  square  decimeter,  114 

per  square  foot,  114 

per  square  inch,  115 

per  square  meter,  114 

per  square  mil,  115 

per  square  millimeter,  115 

-hour,  table,  116 

do.,  denned,  115 

do.  per  cubic  centimeter,  126 

do.  per  gram,  126 

do.  per  pound,  126 

-second,  116 

-turn,  133 

do.,  denned,  132 

do.  per  centimeter,  137 

do.  per  inch,  136 
Amplitude  of  waves,  121 
Amyl  acetate  lamp,  144 
Ancient  lengths,  34 
Angles: 

plane,  89 

do.,  physical,  18 

do.  as  a  suppressed  quantity,  13 

solid,  89 

do.,  physical,  18 

spherical  right,  89 

grade,  91 
Angular: 

acceleration,  88 

do.,  physical,  19 

momentum,  84 

do.,  physical,  19 

velocity,  86 

do.,  physical,  19 

do.  as  a  frequency,  121 

do.,  rate  of  increase  of,  88 
Angstroem  unit,  31 
Anker,     Amsterdam,    Copenhagen, 
Sweden,  55 

Germany,  54 
Annas,  India,  165 
Anomalistic  month,  94 

year,  95 
Apothecary  weights,  table,  59 

defined,  57 

Apparent  power,  denned,  124 
Apparent  resistance,  denned,  98 
Apparent  solar  day,  94 
Applied  volts,  110 
Approximate  numbers,  accuracy,  9 
Approximate  values,  explained,  28 
Ar,  43 

Arabian  lengths,  34 
Ardeb,  Egypt,  172 
Are,  43 

Argentine,  Argentine,  164 
Arish,  Persia,  34 
Arobe,  Paraguay,  173 
Arpent,  France,  Switezrland,  44 
Arratel,  Portugal,  173 
^rroba,  Argentine,  Brazil,  Cuba,  172 

Spain,  61,  173 

Portugal ,  Venezuela,  173 


Arrobas,  Canaries.  55 
Arschine,  Russia,  33    173 
Art,  Sweden,  61* 
Artaba,  Persia,  55 
Artel,  Morocco,  172 
As,  Germany,  60 
•Ass,  Sweden,  61 
Astronomical  day   94 
Atmosphere,  table,  66 

defined,  63 

digit  conversion  table.  67 
Atom,  gram-,  60 
Atomic  weight  of  silver,  125 
Aune,  France,  33 
Austrian  lengths,  33 
Avoirdupois  weights,  table,  59 

defined,  57 
Azumbras,  Spain,  55 


B 

Babylonian  lengths,  34 
Bale,  168 
Bar,  58 
Barie,  64,  66 
Baril,  Argentine,  172 
Barille,  Italy,  55 
Barleycorn,  31 
Barrel,  table,  53 

no  legal  value,  45 

flour,  60 

pork  or  beef,  60 

Malta,  172 

Spain,  173 
Barrique,  France   55 
Bars,  weights  of,  62 
Barye  (see  Barie),  64,  66 
Base  of  logarithms,  171 
Batman,  Persia,  173 
Battery,  voltage  of,  128 
B.A.  unit,  99 

do.  defined,  98 
Becher,  Austria,  55 
Berkovets,  Russia,  173 
Berkowitz,  Russia,  61 
Berri,  Turkey,  34 
Biblical  lengths,  34 
Bloom  ton,  60 
Board  foot,  52 
Board  of  Trade  ohm,  99 

do.  defined,  98 

Board  of  Trade  unit  (energy), 
Boccali,  Rome,  55 
Boisseau,  France,  55 
Bolivar,  Venezuela,  166 
Boliviano,  Bolivia,  164,  166 
Bolt,  32 

Bongkal,  India,  172 
Botschka,  Russia,  55 
Bougie  decimale,  146 

do.  defined,  145 
Bouw,  Sumatra,  173 
Brace io,  Italy,  34 
Brasse,  France,  33 


INDEX. 


177 


Briggs'  logarithms,  171 
Brightness  (light)  table,  148 

do.    physical,  25 
British  Association  unit,  99 

do.  defined,  98 
British  dollar,  China,  164 
British  standard  candle,  146 

do.  defined,  145 
British  thermal  unit,  75 
British  to  U.  S.,  volumes   46 
BTU  (kilowatt-hour),  77  ' 
BTU  (thermal  unit),  75 
Bu,  Japan,  34,  172 
Building  square,  43 
Bundle,  168 
Bushel,  table,  50 

legal,  for  grain,  45 

U.  S.  standard,  45 

to  hectoliters,  digit  table,  51 

salt  (weight),  60 

unusual  values,  53 

weight  of  water,  70 
Butt,  53 

Spain,  173 


Cafoi 


Cable,  32 
Cable's  length,  32 
Cadmium  cell,  108,  110,  111 
Calendar  day,  94 

month,  94 

year,  94 

uorie,  defined,  73 

large,  table,  76 

do.  into  kgl-met.,  digit  table, 77 

do.  per  minute,  81 

small,  table,  75 

do.  into  joules,  digit  table,  77 

do.  per  minute,  80 

do.  per  second,  81 
Calories  into  volts,  129 
Candle,  table    146 

defined,  144 

British  standard,  table,  146 

do.  defined,  145 

do.  hemispherical,  147 

do.  spherical,  147 

English  standard,  table,  146 

do.  defined,  145 

do.  hemispherical,  147 

do.  spherical,  147 

German  paraffine,  table,  146 

do. defined, 145 

hefner,  146 

do.  defined,  144 

do.  hemispherical,  147 

do.  spherical,  147 

hemispherical,  147 

spermaceti,  defined,  145 

spherical,  147 

relation  to  other  units,  146 


Candle  power,  see  Candle 
Candle  per  foot,  148 

per  horse-power,  149 

per  horse-power,  metric,  149 

per  kilo.watt,  149 

per  meter,  148 

per  watt,  149 
Candy,  India,  172 
Cantar,  Malta,  Morocco,  172 

Syria,  Turkey,  173 
Cantara,  Spain,  55 
Cantaro,  Malta,  172 
Caffiri,  Malta,  Sicily,  Tripoli,  Tunis, 

55 

CafBso,  Malta,  172 
Capacity,  electrical,  117 

do.,  C.G.S.  units,  117 

do.  physical  quantity.  23,  24 

do.  inductive,  23 

do.  reactance,  22,  24 

do.  specific  inductive,  23,  24 
Capacity,  magnetic  (permeance), 130 

do.,  physical,  20,  21 

do.,  specific  inductive,  20,  21 
Capacity,  heat,  26 
Capacity,  volumes, cubic, table,  46 

do. ,  digit  conversion  tables,  51 

do., text,  45 

do.,  foreign,  54,  172 
Capillarity,  62 
Carat,  diamond,  59 

Germany,  60 

Amsterdam,      Austria,      Borneo 
France,  Italy,  Lisbon,  Madras 
Spain,  61 
Carcel,  table,  146 

defined, 145 
Carga,  Mexico,  172 
Car-mile,  7 8 
Castellanos,  Spain,  61 
Catrize ,  Spain ,  55 
Catty,  Batavia,  61 

China,  61,  172 

Japan,  61,  172 

Java, 172 

Siam,  173 

Sumatra,  61,  173 
Cent,  Germany,  60 

Haiti,  Netherlands,  165 

U.  S.,  166 

per  foot,  166 

per  ounce,  167 
Centaire,  43 
Centar,  43 
Centare.  43 

Centaro,  Central  America,  172 
Centas,  Germany,  60 
Centavo,  Argentine,  Bolivia,  Chili 
Colombia,  164 

Mexico,  Peru,  165 

Uruguay, 166 
Centen,  Cuba,  164 
Centesimo,  Italy,  165 

Uruguay, 166 
Centi-,  as  prefix,  9 
Centigrade: 

heat  unit,  gram,  75 

do.  kilogram,  76 


178 


INDEX. 


Centigrade  : 

degrees,  reduction  factors,  150 
scale,  to  others,  151-163 
Centigram,  57 
Centiliter,  52 
Centime,  Belgium,  164 
France,  Spain,  165 
Switzerland,  166 
Centimeter,  table,  30 
circular,  42 
inductance,  119 

cubic,  46 

do.  to  cb.  inch,  digit  table,  51 

do.  to  fluid  drams,  digit  table,  51 

do.  to  fluid  ounces,  digit  table,  51 

map  scales,  168 

per  second,  85 

per  second  per  second,  88 

square,  42 

do.  to  sq.  inch,  digit  table,  44 
Centimeter-dyne,  74 

do.  per  second,  80 

-gram,  74 

-gram  per  second,  80 
Centime,  Costa  Rica,  164 
Centistere,  52 
Centner,  Austria,  Sweden,  61 

Germany,  60 

Denmark,  172 

Sweden,  173 
Century,  95 
C.G.S.: 

system  of  units,  text,  11 

unit  current-turn,  133,  134 

do.  per  centimeter,  137 

units,  see  under  respective  names 
Chain,  32 

square,  43 
Chaldron,  54 
Charge  electrical,  115 

do.,  physical,  22,  24 

ionic,  126 

do.  physical,  23,  25 
Charge,  volume,  Candia,  France,  55 
Chemical  weights,  60 
Chetvert,  Russia,  55,  173 
Cheval  vapeur,  81 
Chih,  China,  172 
Cho,  Japan,  34,  44 
Chopine,  France,  54 
Circuit,  kinetic  energy  in,  122 
Circular  unit,  defined,  41 

acre,  44 

centimeter,  42 

foot,  42 

inch,  42 

mil   41 

mil-foot  unit,  102 

millimeter,  41 

Circular  measure  (angles),  89 
Circumference,  89 
Civil  day,  month ,  year,  94 
Clark  cell: 

denned, 108 

e.m.f.  of,  110 

temperature  correction,  111 
Clark  meter,  29 
Coatings,  weights  of,  63 


Coefficient  of: 

expansion,  26 

mutual  induction,  118 

Peltier  effect    23,  25 

self-induction,  table,  119 

do.  defined,  118 

do.  physical,  23,  25 

traction,  78 
Colon,  Costa  Rica,  164 
Committee  meter,  29,  32 
Common  logarithms,  171 
Common  pace,  31 
Compound  names  of  units,  4 
Concrete  temperature  scale,  151-161 
Concrete  vs.  absolute  units,  12 
Concrete  electrical  units,  14 
Condensance,  22,  24 
Condensed  numbers,  8 
Condor,  Chile  Colombia,  164 
Conductance  tables,  104 


physical,  22   24 
C.G.S.  unit  of,  105 


magnetic,  130 
specific,  106 
do.  physical,  22,  24 
Conductivity,  electric,  tables,  107 
text,  106 

Bhysical,  22,  24 
.G.S.  unit  of,  106,  107 

of  copper,  106,  107 

magnetic,  131 

of  mercury,  107 
Conductivity,  heat,  26 
Congresses,  divisions  of  electrical,  14 
Conversion  factors,  tables,   30- 
173 

text,  27 

digit  tables: 

do.  capacities,  51 

do.  energy,  77 

do.  grades,  92 

do.  heat,  77 

do.  inches,  fractions,  mm.,  ft.,  35 

do.  lengths,  39 

do.  power,  82 

do.  pressures,  67 

do.  surfaces,  44 

do.  volumes,  51 

do.  weights,  59 

do.  work,  77 
Coomb,  53 
Copper,  conductivity,  of  106, 107 

resistivity  of,  102,  104 

unit  of  conductivity,  106, 107 
Cord   54 
Cord  foot,  53 
Corde,  Sweden,  33 
Coulomb,  table,  116 

defined,  115 

international,  116 

do.  defined,  115 

true,  116 

per  milligram.  126 
Coupes,  Geneva,  55 
Couple,  physical,  19 
Coyan,  Sarawak,  Siam,  173 
Cross-section  and  resistance,  101 
Cross-section  units,  41 


INDEX. 


179 


Crown,  Austria,  Denmark,  164 

Great  Brit£*i^,  Norway,  Portugal, 

Sweden, 165 
Cuadra,  Argentine,  172 

Paraguay,  Uruguay,  173 
Cuadra  sq.,  Paraguay,  173 
Cubic  centimeter,  46 

to  cb.  inches,  digit  table,  51 

to  fluid  drams,  digit  table,  51 

to  fluid  ounces,  digit  table,  51 

weight  of  water,  70 

per  ampere-hour,  126 
Cubic  decimeter,  48 
Cubic  foot.  49 

to  cb.  meters,  digit  table,  51 

weight  of  water  70 
Cubic  inch,  47 

to  cb.  centimeters,  digit  tables,  51 

weight  of  water,  70 
Cubic  measures ,  45-55 

do.  foreign,  54 
Cubic  meter   51 

to  cb.  feet,  digit  table,  51 

to  cb.  yards,  digit  table,  51 

weight  of  water,  70 
Cubic  millimeter,  52 
Cubic  yard,  50 

to  cb.  meters,  digit  table,  51 

weight  of  water,  70 
Cubit,  31 

ancient,  34 
Currencies,  164 

fluctuating,  166 
Current  (elec.),  table,  113 

text,  112 

physical,  22,  24 

C.G.S.  unit,  112,  113 

density,  114 

do.,  physical,  22,  24 

intensity,  112 

kinetic  energy  of,  122 

strength,  112 

-turn.  C.G.S.  unit  of,  133 
Curvature,  18 

Curvature,  specific,  of  a  surface,  18 
Cycle,  lunisolar,  solar,  95 

frequency,  86,  121 


Daniell  cell,  109,  110 

Day,  94 

apparent  solar,  astronomical,  cal- 
endar, civil,  natural,  nautical, 
sidereal,  solar,  94 

Day's  journey,  ancient,  34 

Deca-,  as  prefix,  9 

Decagram,  59 

Decaliter,  52 


Decameter,  32 
Decare,  44 
Decastere,  54 
Deci-,  as  prefix,  9 
Deciare,  43 
Decigram,  57 
Deciliter,  47 
Decimeter,  30 

cubic,  48 

square,  41 
Decimo,  Chile,  Colombia,  164 

Venezuela,  166 

Decisions  of  electrical  congresses,  14 
Decistere,53 

Decomposition  voltage,  128 
Degree,  angle,  89 

grade,  91 

latitude,  longitude,  32 

per  second,  86 

electrical,  87,89,  121 
Deka,  Germany,  60 
Deka-,  as  prefix,  9 
Dekagram,  59 
Dekaliter,  52 
Dekameter,  32 
Dekastere,  54 
Demi-posson,  France,  54 
Demi-setier,  France,  54 
Denier,  France,  61 
Densities,  mass,  67 

weights  and  volumes  from,  69 

physical,  18 

current,  114 

do.  physical,  24 

flux  (magnetic),  140 

do.  physical,  20,  21 

surface,  electric,  22,  24 
Deposition,  electric .  23,  25 

electrolytic,  126 
Deposits,  electrolytic,  126 

weights  of,  63 

Derivation  of  physical  quantities,  18 
Dessaetine,  Russia,  44 

Spain,  173 

Dimagnetic  bodies,  131 
Diameter,  earth's,  earth's  orbit,  32 
Diameter  to  cross-section,  41 
Decistere,  53 

Dielectric  constant,  23,  24 
Dierno,  Colombia,  164 
Difference  of  potential,  electric,  108 

physical,  22,  24 

magnetic,  132 
Digit,  ancient,  34 
l>igit  conversion  tables: 

capacities,  51 

energy, 77 

grades,  92 

heat.  77 

inches,  fractions,  mm.,  ft.,  35 

lengths,  39 

power,  82 

pressures,  67 

surfaces,  44 

volumes,  51 

weights,  59 

work,  77 
Dime,  U.  S.,  166 


180 


INDEX^ 


Dimensional  formulas,  text,  12 

tables,  18 
Dinero,  Chile,  164 

Peru,  165 

Directive  force,  suspensions,  19 
Discharges  (water),  95 
Displacement,  electric,  22,  24 
Distinction  between  units  and  quan- 
tities, 4 

Doli,  Russia,  61 
Dollar,  British  Honduras,  164 

British  possessions,  164 

China,  164 

Mexico,  165,  166 

Newfoundland,  165 

United  States,  166 

per  mile,  166 

per  pound,  167 

per  ton,  167 

Doubloon,  Chile,  Cuba,  164 
Dozen,  168 
Drachm,  fluid,  52 
Drachm,  weight,  59 
Drachma,  Greece  (money),  165 
Drachm e,  Germany,  60 

Greece  (weight^,  172 
Dragme,  France,  61 
Dram,  fluid,  52 

weight,  59 
Drop,  52 

Dubloon,  Mexico,  165 
Dyne,  83 

-centimeter,  74 

do.  per  second,  80 

per  centimeter,  62 

per  sq.  centimeter,  64 


E 

Eagle,  U.  S.,  166 

Earth,  diameter,  32 

Earth,  diameter  of  orbit,  32 

Earth's  magnetic  field,  140 

Effective  values,  97 

Efficiency  (power),  19 
of  light,  149 

Effort,  tractive,  78 

Egyptian  lengths,  34 

Eimer,  Austria,  Sweden,  55 
Germany,  54 

Electric: 

and  magnetic  units,  96-144 
do.,  interrelations  of,  96 
capacity,  117 
degree,  87,  89,  121 
deposition,  126 
do.,  physical,  23,  25 
displacement,  22,  24 
energy,  tables,  74,  123 
do.,  text,  122 


Electric: 

energy,  C.G.S.  unit,  122,  123 

do.,  physical,  23,  25 

field,  intensity,  22,  24 

inductive  capacity,  18,  23,  24 

power,  tables,  80,  125 

do.,  text,  124 

do.,  physical,  23,  25 

pressure,  108 

quantities,  22.  24 

quantity,  115 

do.,  physical,  22,  24 

stress,  108 

units,  congress  decisions,  14 

units,  relations  to  others,  3 
Electrical,  see  Electric 
Electricity,  quantity,  115 

do.,  physical,  22,  24 
Elect  rochemical  : 

equivalent,  125 

do.,  physical,  23,  25 

do.  of  silver,  125 

energy,  128 

quantities,  23,  25 
Electro-kinetic  inertia,  23,  25 
Electro-kinetic  momentum,  23,  25 
Electrolytic  deposits,  126 
Electrolytic  gas,  126 
Electromagnetic  system,  11 

do.,  units  and  quantities,  20,  22 
Electromagnetic  units,  see  under  re- 
spective names 
Electromotive  force: 

table,  109 

text,  108 

physical,  22,  24 

C.G.S.  unit  of,  109,  110 

at  a  point,  22,  24 

of  Clark  cell,  108,  110 

do.,  temperature  correction,  111 

of  Weston  cells,  108,  110 

do.  temperature  correction,  111 
Electrostatic  system,  11 

do.,  units  and  quantities,  21,  24 
Electrostatic    units,  see    under   re- 
spective names. 
Ell,  32 

Elle,  Austria,  Germany,  33 
Emissivity,  heat,  26 
Energy: 

table,  74 

digit  tables,  77 

text,  72 

physical,  19 

electric,  122 

do.,  physical,  23,  25 

do.,  C.G.S.  unit,  122 

do.,  stored,  123 

kinetic  (electro-),  23,  122 

electrochemical,  128 

magnetic,  20,  21,  143 

rate  of ,  79 

relations  with  torque,  table,  78 

do.,  text,  72 

traction,  78 

do.  vs.  torque,  13,  72,  78 
English  standard  candle,  145,  146 
Entropy,  26 


INDEX. 


181 


Equivalent,  mechanical: 

heat,  3,  26, 72,  171 

light,  147 
Equivalents,  electrochemical,  125 

do.,  physical,  23,  25 
Equivalents  of  units,  30-173 
Erg,  74,  122 

kinetic  energy  of  current,  122, 123 

magnetic  energy,  143 

stored  energy,  122,  123 

per  second,  80 

do.,  electric,  124 

do.,  magnetic,  144 
Errors  of  abbreviated  numbers,  10 
Escudo,  Chile,  164 
Expansion,  temperature,  26 


Factor,  inductance,    23,   118,   120, 
124, 125 

load,  80 

power,  23,79,  124,  125 
Factors,  conversion,  text,  27 

do.,  tables,  30-173 

reduction,  text,  27 

do.,  tables,  30-173 
Fahrenheit  degree  red.  factors,  150 

scale,  to  others,  151-163 

heat  unit,  75 
Fall  of  potential,  108 
Famn,  Sweden,  33 
Fanega    (dry),     Central     America, 
Chile,  Cuba,  Mexico,  Morocco, 
172 

Lisbon,  55 

Spain   55,  173 

Uruguay,  Venezuela,  173 
Fanegada,  Canary  Isles,  Spain,  44 
Farad,  tables,  117 

international,  117 
Faraday's  law,  125 
Fardingdeal,  44 
Farthing,  Great  Britain,  165 
Fathom,  32 

ancient,  34 
Faux,  Switzerland,  44 
Feddan,  Egypt,  172 
Feet,  see  under  Foot 
Feuillette,  France,  54 
Field,  earth's  magnetic,  140 
Field  intensity,  electric,  22,  24 

do,,  magnetic,  134 

do.,  physical,  20,21 
Film  tension,  62 
i  inal  amperes.  113 


Finger,  ancient,  34 
Firkin ,  53 

butter,  60 
Firloi,,  Scotland,  55 
Florin,  Netherlands,  165 
Flow  of  water,  95 
Fluctuating  currencies,  166 
Fluid  dram,  52 

do.  to  cb.  cm.,  digit  table,  51 

scruple,  52 

ounce,  52 

do.  to  cb.  cm.,  digit  table,  51 
Flux: 

light,  table,  149 

do.,  physical,  25 

force,  magnetic,  137 

magnetic,  tables,  138 

do.,  text,  137 

do.,  physical,  20,  21 

do.  density,  140 

do.  density,  physical,  20,  21 

do.  density,  C.G.S.  unit,  140,  141 

do.  from  unit  pole,  138 

-turns,  138 
Florin,  Austria,  164 
Foot: 

tables,  30 

to  inches  and  fractions,  35 

to  meters,  digit  table,  39 

to  millimeters,  digit  table,  35 

board,  52 

builder's,  34 

circular,  42 

cord,  53 

cubic,  table,  49 

do.  to  cb.  m.,  table,  51 

do.  weight  of  water,  70 

foreign:  Austria,  Russia,  33 
Spain,  Italy,  miscellaneous,  34 

mathematic,  34 

pressure  of  water,  65 

rise  per  foot  (grade),  90,  91,  92 

solid  (timber),  53 

square,  table,  42 

do.  to  sq.  meters,  digit  table,  44 

surveyor's,  34 

tradesman's,  34 

water  column,  65 
Foot  per: 

foot  (grade),  91,  92 

100  feet  .(grade),  90,  92 

1000  feet  (grade),  90,  92 

mile  (grade),  90,  91,  92 

minute,  85 

second,  85 

second  per  second,  88 
Foot-: 

-candle,  148 

-grain,  74 

-grain  per  second,  80 

-pound, 74 

do.  per  minute,  80 

do.  per  radian,  78 

do.  per  revolution,  78 

do.  per  second,  80 

do.  to  kilogram-meters,  table,  77 

do.  to  thermal  units,  table,  77 

do.  to  torque,  78 


182 


INDEX. 


Force,  table  and  text,  83 

physical,  18 

center  of  attraction,  18 

directive,  suspensions,  19 

and  length.  62 

and  surface,  63 

per  unit  area,  63 

de  cheval,  81 

flux  of  (magnetic),  137 

lines  of  (magnetic),  137 

do.,  physical,  20,  21 

magnetic,  134 

magnetizing,  134 

do.,  physical,  20,  21 

magnetomotive,  132 

do.,  physical,  20,  21 

tractive,  78 

Foreign  measures,  see  under  specific 
names 

do.,  miscellaneous,  172 
Formulas,  changing  units  of,  6 

temperature  standard  cells,  111 
Fot,  Sweden,  33 
Fot  square,  Sweden,  44 
Fother,  60 
Frail,  Spain,  173 
Franc,  Belgium,  164 

France,  165 

•Switzerland,  166 

per  kilogram,  167 

per  kilometer,  166 

per  meter,  166 

per  metric  ton,  167 
France,  lengths,  33 
Frasco,  Argentine,  Mexico,  172 
Frasila,  Zanzibar    173 
French  horse-power,  81 
Frequency,  general,  86 

electric,  121 

do.,  physical,  23,  25 
Fuder,  Germany,  54 

Luxemburg,  172 
Fun,  Japan,  61 

Functions,  periodically  varying,  97 
Fundamental,  electric  units,  96 

quantities.  18 
Funt,  Russia,  173 
Furlong,  32 
Fuss,  Austria,  GermanyB  Holland, 

Switzerland,  33 
Futtermaassel,  Austria,  55 


G 

Gallon,  liquid,  table,  49 
standard  value,  45 
to  liters,  digit  table,  51 


Gallon,  apothecary,  52 

beer,  52 

British,  49      . 

do.,  standard  value,  46 

dry,  U.  S.,  52 

imperial,  49 

do.,  standard  value,  46 

weight  of  water,  70 

wine,  old  British,  52 
Garnez.  Russia,  55 
Garnice,  Russian  Poland,  173 
Gas,  electrolytic,  126 
Gauss,  141 

defined,  140 

as  magnetizing  force,  137 

do.  defined,  135 

inch-,  141 
Geira,  Portugal,  44 
Geographical  mile,  32 

do.  international,  32 
Geometric  quantities,  18 
German  lengths,  33 

parafnne  candle,  146 

do.  defined,  145 
Gilbert,  133 

defined,  133 

per  centimeter,  136 

per  inch,  136  * 

Gill,  52 

Go,  Japan, 55     . 
Gold  grains,  Spanish,  61 
Gourde,  Haiti,  165 
Grade  (angle),  89 
Grade  (incline).  90 

conversion  table,  92 
Grain,  mass,  weight,  57 

force,  83 

French,  61 

gold,  Spanish,  61 

jeweller's,  59 

volume  of  water,  71 

to  milligrams  digit  table,  59 

per  cubic  inch,  68 

per  inch,  62 
Grain -foot,  74 

per  second,  80 
Grain,  mass,  weight,  58 

force,  83 

volume  of  water,  71 

to  ounce,  digit  table,  59 

per  ampere-hour,  126 

per  centimeter,  62 

per  cubic  centimeter,  69 

per  hour,  127 

per  meter,  62 

per  minute,  127 

per  sq.  centimeter,  64 

per  sq.  decimeter,  64 

per  watt-hour,  126 

-atom,  60 

-centigrade  heat  unit,  75 

-centimeter,  74 

do.  per  second,  80 

-molecule,  defined,  60 

do.  gas  volume,  53 
Gran.  Germany,  60 
Gravitation  constant,  physical,  18 

do.  defined,  87 


INDEX. 


183 


Gravitational  units,  relations,  3 
Gravity,  88,  171 

mean  value,  87 

as  a  relation,  3 
Great  gross,  168 
Grecian  lengths,  34 
Gregorian  year,  95 
Grivinik,  Russia,  165 
Gros,  France,  61 
Gross,  168 
Gross  ton ,  58 

see  also  Ton,  long 

displacement  of  water,  54 
Guinea,  Great  Britain,  165 


H 

Hairsbreadth,  31 

Hand,  31 

Harcourt  pentane  lamp,  146 

do.  defined,  145 
Heaped  bushel,  53 
Heat,  tables,  74 

digit  tables,  77 

text,  72 

physical,  26 

of  combination  into  volts,  129 

latent,  26 

mechanical  equivalent,  171 

do.  defined,  72 

do.,  physical,  26 

do.  as  a  relation,  3 

specific,  capacity,  26 

specific,  of  water,  171 

do.  as  a  relation,  3 

unit,  75,  76 

do.  defined,  73- 

do.  digit  tables,  77 

do.  relations  to  others,  3 
Hebrew  lengths,  34 
Hecta-,  as  prefix,  9 
Hectare,  43 

to  acres,  digit  table,  44 
Hectogram,  60 
Hectoliter,  50 

to  bushels,  digit  table,  51 

weight  of  water,  70 
Hectometer,  32 
Hectostere,  54 
Hefner  or  hefner  unit,  146 

defined,  144 

hemispherical,  147 

spherical,  147 
Hekto  ,  as  prefix,  9 
Hektoliter,  50 
Hektostere,  54 
Heller,  Austria,  164 
Hellergewicht,  Germany,  60 


Hemisphere,  89 
Hemispherical  candle,  147 
Hemispherical  hefner,  147 
Henry,  119 

defined,  118 

relations  to  other  units,  119 
Hogshead,  53 
Holland  lengths,  33 
Horse- power,  table,  81 

metric,  see  under  Metric 

to  metric  hp.,  digit  table  82 

to  kilowatts,  digit  table,  82 

-minute,  76 

do.,  metric,  76 

-second,  75 

do.,  metric,  75 

-hour,  77 

do.,  me  trie,  77 

do.  per  kilogram,  126 

do.  per  minute,  82 

do.  per  pound,  126 

do.  per  second,  82 
Hour  (solar),  94 

sidereal  94 

Hyperbolic  logarithms,  171 
Hyphen,  in  names  of  units,  4 
Hundredweight,  58 


Illumination,  148 

physical,  25 
Immissivity,  heat,  26 
Impact,  19 
Impedance,  99 

defined,  98 

physical,  22,  24 
Imperial,  Russia,  165 
Imperial  gallon,  49 

defined,  46 

Incandescent  lamp  standards,  145 
Inch,  table,  30 

fractions,  mm..,  and  feet,  tables,  35 

tp  millimeters,  digit  tables,  39 

circular,  42 

cubic,  47 

do.  to  cb.  cm.,  digit  table,  51 

do.,  weight  of  water,  70 

square,  42 

do.  to  sq.  cm.,  digit  table,  44 

gauss,  141 

on  map,  168 

mercury  column,  65 

per  mile,  90,  92 
Inclines,  90 
Induced  volts,  110 


184 


INDEX. 


Inductance,  table,  119 

text,  118 

C.G.S.  unit,  119 

physical,  23,  25 

mutual,  25 
Inductance  factor,  defined,  118,  124 

do.,  physical,  23 

do.,  values,  120,  125 
Induction,  118 

capacity,  24 

magnetic,  140, 

do.,  physical,  20,  21 

mutual,  23,  118 

self,  23,  118 
Inductive  capacity,  electric,  23 

do.,  relation  to  permeability,  14 

do.,  suppressed  factor,  125  14 

do.,  specific,  20,  21,  23 
Inertia,  physical,  19 

electro-kinetic,  23,  25 

moment  of,  84 
Intensity  of: 

attraction,  18 

electric  field,  22,  24 

light,  144 

do.,  physical,  25 

magnetic  field,  134 

do.,  physical,  20,  21 

magnetization,  142 

do.,  C.G.S.  unit  of,  142 

do.,  physical,  20,  21 

stress,  physical,  19 
International : 

ampere,  113 

do.  defined,  112 

coulomb,  116 

do.  defined,  115 

farad,  117 

meter,  defined,  29 

mile,  32 

nautical  mile,  32 

ohm,  99 

do.  defined,  98 

volt,  110 

do.  defined,  108 
Inter-relation  of  units,  1 
Introduction,  1-26 
Ion    125,  128,  129 

energy  of,  129 
Ionic  charge,  126 

physical,  23,  25 
Irrigation  units,  95 
Italian  lengths,  34 


Japanese  lengths,  34 
Jeweller's  grain,  59 
Jo,  Japan,  34 
Joch,  Austria.  44 


Joule: 

table,  74,  123 
defined,  122     • 
to  calories,  digit  table,  77 
electro-chemical  energy.  128 
kinetic  energy  of  current,  122,  123 
magnetic  energy,  143 
per  cycle,  123 
per  second,  80 
Julian  year,  94 


Kanne,  Sweden,  55 
Kapp  line,  138 
Kater,  value  of  meter,  29 
Ken,  Japan,  34,  172 

square,  Japan,  44 
Kern,  Germany,  60 
Klafter,  Austria,  33,  44 

Germany,  54 

Russia,  173 
Kilo,  58 

Kilo-,  as  prefix,  9 
Kiloampere,  113 
Kilodyne,  83 
Kilogauss,  141 
Kilogram: 

table,  58 

defined,  56 

relation  to  liter,  56 

force,  83 

to  pounds,  digit  table,  59 

volume  of  water,  71 

per  cubic  centimeter,  69 

per  cubic  meter;  68 

per  day,  127 

per  hectoliter,  68 

per  horse -power-hour,  126 

per  hour,  127 

per  kilometer,  62 

per  kilowatt  hour,  126 

per  liter,  69 

per  meter,  63 

per  sq.  centimeter,  66 

do.  to  atmospheres,  digit  table,  67 

do.  to  Ibs.  per  sq.  in.,  table,  67 

per  square  meter,  64 

per  square  millimeter,  66 

per  ton,  78 

per  year,  126 

-centigrade  heat  unit,  76 

-kilometer,  76 

do.  per  minute,  81 

-meter,  75 

do.  to  foot-pounds,  digit  table, 77 

do.  to  large  calories,  digit  table, 77 

do.  per  minute,  80 

do.  per  second,  81 

-molecule,  60 
Kilo  joule,  123 
Kiloliter,  54 


INDEX. 


185 


Kilometer,  table,  30 

to  miles,  digit  table,  39 

square,  43 

per  hour,  85 

do.  per  minute,  88 

do.  per  second,  88 

per  minute,  86 
Kilo  volt,  110 
Kilowatt,  table,  82,  125 

denned,  124 

to  horse -powers,  digit  table,  82 

do.,  metric,  digit  table,  82 

-hour,  table,  77,  123 

defined,  122 

per  kilogram,  126  . 

per  minute,  82 

per  pound,  126 

per  second,  82 

-minute,  76 

-second,  75 
Kin,  Japan,  61 
Kine,  85 
Kinetic  energy  of  a  current,  122 

do.,  physical,  23 
Kinetic  inertia,  electro-,  23 
Kinetic  momentum,  electro-,  23 
Kislos,  Alexandria,  Constantinople, 

Smyrna,  55 
Knot,  31 

telegraph,  British   32 

per  hour,  85 
Koku,  Japan,  55,  172 
Kopek,  Russia,  165 
Koppa,  Sweden  ,,55 
Korn,  Germany,  60 

Sweden,  61 
Korree,  Russia,  173 
Kran,  Persia,  165,  166 
Kreutzer,  Austria,  164 
Kruschky,  Russia,  55 
K unk as,  Russia,  55 
Kvintin,  Sweden,  61 
Kwan,  Japan,  61 


Lachter,  Germany,  33 
Lamps,  standard,  145 
Last,  54 
Last,  Belgium,  Holland,  172 

Russian  Poland,  Spain,  173 
Latent  heat,  26 
Latitude,  degree  of,  32 
League,  32 
League,  France,  33  ' 

Paraguay,  173 

Spain,  34 

miscellaneous  foreign,  34 


Legal  ohm,  99 

do.  defined,  98 

volt,  109 

year,  95 

see  also  Standards 
Legua,  32 
Lengths,  table,  30 

text,  29 

physical,  18 

British  standards,  29 

foreign,  33 

fundamental  standards,  29 

systeme  ancien,  France,  33 

systeme  usuel,  France,  33 

U.  S.  standards,  29 

and  forces,  62 

and  masses,  62 

and  money,  166 

and  weights,  62 
Lepta,  Greece,  165 
Li,  China,  34,  172 
Libra,  Argentine,  Central  America, 
Chile,  Cuba,  Mexico,  172 

Peru,  165,  173 

Spain,  61,  173 

Portugal,    Uruguay,    Venezuela, 

173 

Liespund,  Sweden,  61 
Lieue,  France,  33 

marine,  France,  33 

moyenne,  France,  33 
Li  ghlt,  144-149 

physical,  25 

brightness  of  source,  148 

candle  power,  144 

dimensional  formulas,  25 

efficiency,  149 

flux  of,  147 

illumination,  148 

intensity  of,  144 

mechanical  equivalent,  147 

quantity  of,  149 

radiant,  as  power,  3,  25,  147 

standards  of,  146 

do.  defined,  144 

velocity  of,  86 

do.  as  a  relation,  11, 14,  21,  24,  25. 
96 

units,  144-149 

do.,  absolute,  3,  145,  146 

do.,  relations  to  others,  3 
Ligne,  France,  33 
Linear  acceleration ,  87 

do.,  physical,  19 
Linear  velocity,  85 

do.,  physical,  19 
Line,  31 
Lines  of  force  (magnetic),  137 

do.   physical,  20,  21 

dp.  per  unit  cross-section,  140 
Linie,  Austria,   Germany,  Sweden, 

Switzerland,  33 
Link,  31 
Lira,  Italy,  165 
Liter,  table,  48 

defined,  45 

relation  to  kilogram ,  45 

standard,  defined,  45 


186 


INDEX. 


Liter,  weight  of  water,  70 

to  gallons,  digit  table,  51 

to  quarts,  digit  table,  51 
Litron,  France,  55 
Livre,  France,  61 

Greece,  Guiana,  172 
Load  factor,  80 
Lod,  Sweden,  61 
Logarithms,  accuracy  of,  10 

bases,  171 

conversions  of,  171 

systems  of,  171 
Long  ton,  58 

do.,  see  also  Ton,  long 
Longitude,  degree  of,  32 
Loop,  Russia,  55 
Loth,  Germany,  60 

Austria,  Russia,  Switzerland,  61 
Louis,  France,  165 
Lumen,  147 
Lumen-hour,  149 
Lunar  month,  year,  94 
Lunisolar  cycle,  95 
Lux, 148 


M 

Maass,  Austria,  55 
Maassel,  Austria,  55 
Magnetic: 

and  electric  units,  96-144 

do.,  dimensional  formulas,  20 

do.,  introductory  text,  96 

do.,  interrelations,  96 

calculations,  136 

capacity,  130 

do.,  physical,  20,  21 

conductance,  130 

conductivity,  131 

energy,  143 

do.,  physical,  20,  21 

field,  137 

do.  defined,  138 

force,  134 

flux,  tables,  138 

do.,  text,  137 

do.,  physical,  20,  21 

do.,  C.G.S.  unit,  138 

flux  density,  140 

do.,  physical,  20,  21 

do.,  C.G.S.  unit,  140,  141 

flux  from  unit  pole,  138 

flux-turns,  138 

induction,  140 

do.,  physical,  20,  21 

do.,  C.G.S.  unit,  141 

lines  of  force,  137 

do.  physical,  20,  21 

do.  per  unit  cross-section,  140 

moment,  142 

do.,  physical/20,  21 

do.,  C.G.S.  unit,  142 


Magnetic; 

permeability,  131 

do.,  physical,  18,  20,  21 

do.,  suppressed  factor,  12,  14 

permeance,  130 

do.,  C.G.S.  unit,  131 

pole,  unit,  138 

potential,  132 

do.,  physical,  20,  21 

power,  144 

do.,  physical,  20,  21 

pressure,  132 

quantities,  physical,  20,  21 

reactance,  22,  24 

reluctance,  129 

do.,  physical,  20,  21 

do.,  C.G.S.  unit,  129 

reluctivity,  130 

do.,  physical,  20,  21 

resistance,  129 

do.,  physical,  20,  21 

resistivity,  130 

susceptibility,  132 

do.,  physical,  20,  21 

units,  129-144 

do.,  congress  decisions,  14 

do.,  general,  96 

work,  143 

Magnetisation,  see  Magnetization 
Magnetization,  intensity,  142 

do.,  physical,  20,  21 

do.,  C.G.S.  unit,  142 
Magnetizing  force,  tables,  137 

text,  134 

physical,  20,  21 

C.G.S.  unit,  136 

do.  defined,  134 

units,  defined,  134 
Magnetomotive  force,  132 

physical,  20,  21 

C.G.S.  unit,  133 

per  centimeter,  134 
Malter,  Germany,  54 
Manzana,     Costa-Rica,    Nicaragua 
172 

Salvador,  173 
Marc,  France,  61 

Bolivia,  172 
Marco,  Spain,  61 
Mark,  Finland,  164 
Mark  (money),  Germany,  165 

per  meter,  166 

per  metric  ton,  167 

per  kilogram,  167 

per  kilometer,  166 

(weight)  Germany,  60 

do.,  Sweden,  61 
Masses: 

tables,  57 

digit  tables,  59 

text,  56 

foreign,  60,  172 

standards,  defined,  50 

physical,  18 

water,  69 

and  lengths,  62 

and  surfaces,  63 

and  volumes,  67 


INDEX. 


187 


Materials,  weight  of ,67 
Mattari,  Tripoli,  Tunis,  55 
Maund,  India,  172 

Mocha,  61 

Maximum  values,  97 
Maxwell,  138 

per  square  centimeter,  141 

per  square  inch,  141 
Mean  solar  time,  93 
Mean  values,  97 
Mean  watts,  125 
Measures,  see  under  respective  names 

apothecary,  text,  45 

dry,  text,  45 

liquid,  text,  45 
Mechanical  equivalent  of  heat ,  17 1 

do.  denned, 72 

do.  as  a  relation,  3 

do.,  physical,  26 

Mechanical  equivalent  of  light,  147 
Mechanical  quantities,  18 
Medimni,  Greece,  55 
Mega-,  as  prefix,  9 
Megabarie,  66 
Megadyne,  83 

per  square  centimeter,  66 

per  square  meter,  64 
Megamho,  105 
Mega  volt,  110 
Megohm,  99 

cubic  centimeter,  unit,  103 
Meile,  Austria,  Germany,  Sweden,  33 
Mercury,  conductivity,  107 

density,  63 

pressures,  63 

do.,  inch  of,  65 

do.,  millimeter  of,  65 

specific  gravity,  63 

resistivity,  102,  104 

unit,  conductivity,  107 
Meter,  table,  30 

standard,  29 

to  feet,  digit  table,  39 

to  yards,  digit  table,  39 

Clark  value,  29 

committee,  29,  32 

cubic,  51 

do.  to  cb.  ft.,  table,  51 

do.  to  cb.  yds.,  table,  51 

international,  29 

in  wave  lengths,  32 

Kater  value,  29 

legal  standard,  29 

older  values,  29 

Pratt  &  Whitney  Co.,  31 

square,  43 

do.  to  sq.  feet,  table,  44 

do.  to  sq.  yards,  table,  44 

standard,  29 

water  column,  65 

per  minute,  85 

per  second   85 

per  second  per  second,  88 

-candle,  148 

-kilogram,  energy,  75 

do.,  torque,  78 

millimeter  unit,  102 
Metercentner,  Austria,  61 


Metric  horse-power,  81 

to  horse-power,  digit  table,  82 
to  kilowatt,  digit  table,  82 
-hour,  77 

do.  per  minute,  82 
do.  per  second,  82 
-minute,  76 
-second,  75 

Metric  system,  prefixes,  9 
Metric  ton,  58 

per  square  meter,  65 

per  year,  127 
Metze,  Austria,  55 

Germany,  54 
Mho,  105 

cubic  centimeter  unit,  107 

do.  defined,  106 
Micro-,  as  prefix,  9 
Microampere,  113 
Microcoulomb,  116 
Microdyne,  83 
Microfarad,  117 
Microhenry,  119 
Microhm,  99 

cubic  centimeter  unit,  103 

per  cubic  centimeter,  103 

square  cm.  per  cm.,  103 

do.  defined,  102 
Microjoule,  123 
Micro-meter.  31 
Micro-millimeter,  31 
Micron,  31 
Microne,  31 
Microvolt.  109 
Microwatt,  125 
Mil,  30 

circular,  41 

square,  41 

Denmark,  172 
Mil-foot  unit,  102,  103 
Mile,  table,  31 

to  kilometers,  digit  table,  39 

international  geographical,  32 

international  nautical,  32 

geographical,  32 

nautical,  31 

square,  43 

statute,  see  Mile 

telegraph,  34 

German,  Holland,  Swiss,  33 

Netherlands,     Italy,     miscellane 
ous,  34 

per  foot,  91,  92 

per  hour,  85 

do.  per  minute,  88 

do.  per  second,  88 

per  minute,  86 

-pound,  76 

do.  per  hour,  81 

do.  per  minute,  81 

-ton,  78 

Military  pace,  31 
Milla,  Honduras,  Nicaragua,  172 
Millarium,  ancient,  34 
Millennium,  95 
Milli-,  as  prefix,  9 
Milliampere,  113 
Milliare.  43 


188 


INDEX, 


Millier,  58 
Millier,  France,  61 
Milligram ,  mass,  57 

force,  83 

to  grains,  digit  table,  59 

per  coulomb,  126 

per  millimeter,  62 

£e    second,  127 
ihenry,  119 
Milliliter,  46 
Millimeter,  30 

to  fractions  of  inch,  digit  table,  35 

to  inches,  digit  table,  39 

circular,  41 

cubic,  52 

mercury  column,  65 

square,  41 

per  meter,  90,  92 
Milli-micron,  31 
Millimol,  60 
Millistere,  52 
Millivolt,  109 
Milreis,  Brazil,  164 

Portugal,  165 
Mina,  Genoa,  55 
Miner's  inch,  95 
Miner's  pound,  Swedish,  61 
Minim,  52 
Mint  pound,  60 
Minute,  time,  94 

sidereal,  94 

angle,  89 
Miria-,  see  myria 
Miriameter,  32 

square,  44 

Modulus  of  elasticity,  19 
Modulus  of  logarithms,  171 
Moggi,  Milan,  55 
Moggia,  Naples,  44 
Mol,  60 
Mole,  60 
Molecule,  52 

gram,  kilogram,  60 

average  velocity  of,  86 
Moment,  72 

torque,  72-78 

physical,  19 

of  inertia,  84 

do.,  physical,  19 

momentum,  84 

do.,  physical,  19 

magnetic,  142 

do.,  physical,  20,  21 

per  unit  volume,  142 
Momentum  79 

physical,  19 

angular,  84 

electro-kinetic,  23,  25 

moments  of,  84 
Momme,  Japan,  61 
Money,  164 

fluctuating,  166 

and  length,  166 

and  weight,  167 
Monovalent  ions,  129 
Month,  anomalistic,  calendar,  civil, 

lunar,  sidereal,  synodic,  94 
Morgen,  Germany,  44 


Motion,  quantity  of,  19 
Mudde,  Amsterdam,  55 
Muid,  France,  54 
Mut  or  Muth,  Austria,  55 
Mutual  inductance,  118 

do.,  physical,  23,  25 

induction,  118 

do.,  physical,  23,  25 
Myria-,  as  prefix,  9 
Myriadyne,  83 
Myriag/am,  60 
Myrialiter,  54 
Myriameter,  32 

square,  44 
Myrioliter,  54 


N 

Nahud  cubit,  ancient,  34 

Nail,  31 

Names  of  units,  compound,  4 

Naperian  logarithms,  171 

Napoleon,  France,  165 

National  Bur.  Standards  ampere,  113 

National  prototypes,  29 

Natural  day,  94 

Natural  logarithms,  171 

Natural  year,  95 

Nautical  day,  94 

Nautical  mile,  31 

defined   29 

international,  32 
Net  ton,  58 

do.,  see  also  Ton,  short 
Noeud,  France,  33 
Numbers,  accuracy  of,  9 
Numbers,  condensed,  8 
Numbers,  useful,  170 


O 

Oer,  Denmark,  164 
Norway,  Sweden,  165 

Oersted,  129 

Ohm,  table,  99 
text,  98 

Board  of  Trade,  98,  99 
international,  98,  99 
legal,  98,  99 


INDEX. 


189 


Ohm,  true,  98,  99 

Reichsanstalt,  99 

circular-mil,  foot  unit,  102,  103 

circular-mil,  per  foot,  102,  103 

circular-millimeter,    meter    unit, 
103 

circular  millimeter  per  meter,  103 

cubic  centimeter  unit,  102,  103 

sq.  cm.  per  cm.,  103 

sq.  mil,  foot  unit,  102,  103 

sq.  mil  per  foot,  102,  103 

sq.  mm.,  met.,  unit,  102,  103 

sq.  millimeter  per  meter,  102,  103 

per  cubic  centimeter,  103 

per  foot,  101 

do.  per  circular  mil,  102,  103 

do.  per  mil  diameter,  103 

do.  per  square  mil,  102,  103 

per  kilometer,  101 

per  meter,  101 

do.  per  circular  millimeter,  103 

do.  per  sq.  millimeter   102,  103 

per  mile,  101 

-centimeter  square,  101 

-centimeter  unit,  102 

-inches  square,  101 

(volume),  Germany,  54 
Ohmic  resistance,  98 
Oke,  oriental,  61 

Egypt,  Greece,  Hungary,  172 

Turkey,  173 
Once,  France,  61 

Mexico,  165 
Orbit,  earth's,  32 
Orne,  Austria,  55 
Osmini,  Russia,  55 
Ounce  (av.),  mass,  58 

force,  83 

to  grams,  digit  table,  59 

volume  of  water,  71 

apothecary,  59 

fluid,  52 

Troy,  59 

Troy,  silk,  59 

per  hour,  127 

per  minute,  127 
Oxhoft,  Germany,  54 


Pace,  31 

common,  31 

military,  31 
Pajok,  Russia,  55 
Palm,  31 

ancient,  34 

Palmo,  Italy,  Spain,  34 
Paper  measure,  168 
Para,  Egypt,  164 

Turkey,  166 


Paraffin  candle,  146 

denned,  145 

Paramagnetic  bodies,  131 
Parasang,  Persia,  34 
Passus,  ancient,  34 
Peck,  49 

weight  of  water,  70 
Penni,  Finland,  164 
Penny,  Great  Britain,  165 
per  foot,  166 
per  ounce,  167 
Pennyweight,  Troy,  59 
Pentane  lamp,  146 

defined,  145 
Peltier  effect,  23,  25 
Per,  in  names  of  units,  4 
Percent  (grade),  90-92 
Percentage,  defined,  7 
Per  mil  (grade),  90,  92 
Perch,  length,  32 
masonry,  54 
square,  43 

Perche,  France,  33,  44 
Period,  86,  121 

physical,  23,  25 

Periodically-varying  functions,  97 
Periodicity,  86,  121 
Permeability,  magnetic,  131 
physical,  20,  21 

relation  to  inductive  capacity,  14 
suppressed,  factor,  12,  14 
Permeance,  magnetic,  130 
do.,  physical,  20,  21 
specific,  131 
Pes,  ancient,  34 
Peseta,  Spain,  165 
Peso,  Argentine,  Chile,  Cuba,  164 
Colombia,  164,  166 
Guatemala,  Honduras,  Nicaragua, 

Mexico,  Salvador,  165 
Central  America,  Uruguay,  166 
Pezza,  Rome,  44 
Pfennig  (weight),  Austria,  61 

(money),  Germany,  165 
Pfenniggewicht,  Germany,  60 
Pferdekraft,  81 
Pfund,  Germany,  60 

Austria,  Russia,  Switzerland,  61 
Physical  quantities,  table,  17 
derivation  of,  18 
dimensional  formulas  of,  18 
symbols  of,  18 
Photometric  quantities,  25 
units,  144-149 
do.,  congress  decisions,  14 
Pi  (?r),  functions  of,  169 

do.  as  an  angle,  89 
Piaster,  Egypt,  164 

Turkey,  166 
Piastre,  Mexico,  165 
Pic,  Egypt,  172 
Picul,      Borneo,     Celebes,     China, 

Japan,  Java,  172 
Philippine  Islands,  173 
Pie,  Argentine,  172 

Spain,  173 
Pied,  France,  33 
Pietak,  Russia,  165 


190 


INDEX. 


Piff,  Austria,  55 
Pig,  metal,  60 
Pik,  Turkey,  173 
Pint,  47 

weight  of  water,  70 

apothecary,  52 

Scotland,  55 
Pinte,  France,  54 

Genoa,  55 
Pipe,  53 

Pistareen,  Spain,  165 
Plane  angle,  89 

physical,  18 
Platinum  standard,  146 

denned,  145 

Poids  de  Marc,  France,  61 
Point,  31 

France,  33" 
Pole,  length,  32 

square,  43 

(magnetic)  strength,  137 

do.,  physical,  20,  21 

do.  per  unit  cross-section,  142 

do.  unit,  138 
Poltinnik,  Russia,  165 
Poncelet,  82 
Pood,  Russia,  173 
Posson,  France,  54 
Pot,  France,  54 
Potential,  electric: 

absolute,  108 

electromotive  force,  108 

physical,  22,  24 

vector,  22,  24 
Potential,  magnetic,  132 

do.,  physical,  20,  21 
Pottle,  British,  52 
Pouce,  France,  33 
Pound  (mass),  table,  58 

denned,  56 

force,  83 

to  kilograms,  digit  table,  59 

volume  of  water,  71 

apothecary,  60 

(silk),  Lyons,  France,  61 

miner's,  Sweden,  61 

mint,  60 

Troy,  60 

Troy,  silk,  60 

Amsterdam,  Austria,  Italy,  Rot- 
terdam, Russia,  Spain,  miscel- 
laneous, 61 

sterling  (money),  Great  Britain, 
India,  165 

Egypt,  164 
Pound  per: 

ampere-hour,  126 

bushel,  68 

cubic  foot,  68 

cubic  inch,  69 

cubic  yard,  68 

day,  127 

foot,  63 

gallon,  68 

horse-power  hour,  126 

hour,  127 

inch,  63 

kilowatt-hour,  126 


Pound  per: 

mile,  62 

quart,  68 

square  foot,  64 

square  inch,  65 

do.  to  atmospheres,  table,  67 

do.  to  kg.  per  sq.  cm.  table,  67 

ton,  78 

yard,  62 

year,  126 
Pound- : 

-Fahrenheit  heat  unit,  75 

-foot  revolution,  78 

-centigrade  heat  unit,  75 

do.  per  minute,  81 

-foot,  energy,  74 

do.,  torque,  78 

-foot-radian,  78 
Poundal,  83 

per  inch,  63 

per  sq.  foot,  64 

per  sq.  inch,  65 
Pous,  ancient,  34 
Power,  tables,  80-82 

digit  tables,  82 

text,  79 

physical,  19 

C.G.S.  unit,  125 

apparent,  defined,  124 

electric,  124 

do.,  physical,  23,  25 

magnetic,  144 

do.,  physical,  20,  21 

per  candle-power,  149 

radiant  light,  3,  25,  147 
Power  factor,  defined,  79,  124 

do.,  physical,  23 

do.  values,  125 
Practical  units,  12 
Prefixes  in  metric  system,  9 
Pressures,  tables,  64 

digit  tables,  67 

text,  63 

physical,  19 

water,  mercury  and  atmosphere, 
63 

electrical,  108 

magnetic,  132 
Prototypes,  national,  29 
Pud,  Russia,  61 
Puncheon,  53 
Pund,  Sweden,  173 
Pyr,  146 

defined,  145 
it  as  an  angle,  89 
it,  useful  functions  of,  169 


Q 

^uad,  119 
Quadrant,  angle,  89 
inductance,  119 


INDEX. 


191 


Quantities  and  units: 

distinction  between,  4 

table  of,  physical,  17 
Quantity: 

electrical,  115 

do.,  C.G.S.  unit,  116 

do.,  physical,  22,  24 

light,  149 

do.,  physical,  25 

motion.  19 
Quart,  48 

to  liters,  digit  table,  51 

weight  of  water,  70 

Germany,  54 
Quart  ant,  France,  54 
Quartellos,  Spain,  55 
Quarter,  length,  31 

volume,  53 

weight,  60 
Quarti,  Rome,  55 
Quentchen,  Austria,  61 

Germany,  60 
Quintal,  60 

Argentine,  Brazil,  Chile,  Greece, 
Mexico,  Newfoundland,  Para- 
guay, Peru,  Syria,  173 

Spain,  61 

Quinteau,  France,  61 
Quire,  168 


Radian,  89 

per  minute,  86 

per  second,  86 

Radiant  light  as  power,  3,  25, 147 
Rails,  weights  of,  62 
Railway  time,  93 
Rate  of: 

change  of  amperes  per  sec.,  113 

doing  work,  79 

energy,  79 

heat  production,  26 

increase  in  velocity,  87-88 

increase  in  angular  velocity,  88 
Ratios,  described,  7 
Reactance,  98,  100 

physical,  22,  24 

capacity,  22,  24 

magnetic,  22,  24 

units  and  relations,  99-100 
Ream,  168 
Reaumur  degrees,  reduction,  150 

scale  to  others,  151-157 
Rebele,  Alexandria,  55 
Reduction  factors,  tables,  30-173 

do-,  text,  27 
Reducing  units  in  formulas,  6 


Register  ton,  shipping,  54 
Reichsanstalt,  ampere  of,  113 

ohm  of,  99 

volt  of,  110 
Reis,  Brazil,  164 

Portugal,  165 
Relations: 

of  units,  tables,  30-173 

do.,  text,  1 

do.,  elec.  and  mag.,  96 

between  quantities,  13 

physical  quantities,  17 
Reluctance,  magnetic,  129 

do.,  physical,  20,  21 

specific,  130 

do.,  physical,  20,  21 
Reluctivity,  magnetic,  130 

do.,  physical,  20,  21 
Resilience,  19 
Resistance,  electric: 

table,  99 

text,  97 

physical,  22,  24 

C.G.S.  unit,  99 

specific,  table,  103 

do.,  text,  101 

do.,  physical.  22 

and  cross-section,  101 

and  length,  101 

units  and  relations,  99 
Resistance,  magnetic,  129 

do.,  physical,  20,  21 

do.,  specific,  130 

traction,  78 
Resistivity,  table,  103 

text,  101  . 

physical,  22,  24 

C.G.S.  unit,  102,  103 

copper,  102,  104 

mercury,  102,  104 

magnetic,  130 
Revolution,  89 

per  hour,  86 

per  minute,  86 

per  minute  per  minute,  88 

per  minute  per  second,  88 

per  second,  86 
'per  second  per  second,  88 
Rhineland  foot,  33 
Ri,  Japan,  34 
Richtpfennig,  Germany,  60 
Right  angle,  89 

do.,  spherical,  89 
Rod,  32 

Rod,  square,  43 
Rod,  volume,  54 
Roman  lengths,  34 
Rood,  44 

Roquille,  France,  54 
Rotary  speeds,  86 
Rottle,  Palestine,  172 

Syria,  173 

Rottoli,  oriental,  Naples,  61 
Royal  cubit,  ancient,  34 
Rubbio,  Rome,  55 
Ruble,  Russia,  165 
Rupee,  Ceylon,  164 

India,  165 


192 


INDEX. 


Russian  lengths,  33 

Ruthe,  Austria,  Germany,  Holland, 

Sweden,  33 
Ruthe  sq.,  Germany,  44 


3 

Sacco,  leghorn,  55 
Sach,  Rotterdam,  55 
Sack,  53 

Amsterdam,  55 
Sacred  cubit,  ancient,  34 
Salm,  Malta,  55,  172 
Salme,  Sicily,  55 
Saschehn,  Russia,  33 

square,  Russia,  44 
Scale  of  maps  and  drawings,  168 
Schachtruthe,  Germany,  54 
Schalpfund,  Sweden,  61 
Scheffel,  Germany,  54 
Schiffslast,  Germany,  60 
Schiffspund,  Sweden,  61 
Schiffstonne,  Austria,  61 
Score,  168 
Scruple,  apothecary,  59 

fluid,  52 

Scrupule,  Fi»nce,  61 
Se,  Japan, 44, 172 
Sea  mile,  32 
Secchio,  Venice,  55 
Sec-ohm,  119 
Second,  angle,  89 

length,  31 

time,  94 

sidereal,  94 
Section,  of  land,  44 
Seer,  Bengal,  61 

India,  172 
Seidel,  Austria,  55 
Seki,  Japan,  55 
Self -inductance,  118 
Self-induction,  118 

physical,  23,  25 
Semi-circumference,  89 
Sen,  Japan,  165 
Setier,  Geneva,  55 

France,  54 

Shaku,  Japan,  34,  172 
Sheets,  weights  of,  63 
Shilling,  Great  Britain,  165 

per  mile,  166 

per  pound,  167 

per  ton,  167 
Sho,  Japan,  55,  172 
Short  ton,  58 

do.,  see  also  Ton,  short 
Sidereal  day,  hour,  minute,  month, 
second,  year,  94 

time,  defined,  93 


Siemens-unit,  99 

defined,  98 
Silver,  atomic  weight,  125 

electrochemical  equiv.,  125 
Sine,  grades,  91,  92 
Skalpund,  Sweden,  61 
Skilling,  Norway,  165 
Skrupel,  Germany,  60 
Slopes,  90 
Sol,  Peru,  165 
Solar  cycle,  95 
Solar  time,  93 
Solid  angle,  89 

physical,  18 
Solid  yard,  54 
Solotnick,  Russia,  61 
Sovereign,  Great  Britain,  165 
Sou,  France,  165 
Span,  31 

ancient,  34 
Spanish  lengths,  34 
Specific: 

conductance,  106 

do.,  physical,  22,  24 

gravity,  67 

do.,  physical,  18 

do.,  weights  and  volumes  from,  69 

heat,  26 

do.  of  electricity,  23,  25 

do.  of  water,  72,  171 

do.  as  a  relation,  3 

inductive  capacity,  elec.,  23,  24 

do.,  magnetic,  20,  21 

magnetic  reluctance,  130 

magnetic  resistance,  130 

permeance,  131 

reluctance,  130 

do.,  physical,  20,  21 

resistance,  table,  103 

do.,  text,  101 

do.,  physical,  22,  24 
Speeds,  85 

rotary,  86 

Spermaceti  candle,  145 
Sphere,  89 
Spherical  candle,  147 

hefner,  147 

right  angle,  89 
Spon,  Sweden,  55 
Square: 

building,  43 

centimeter,  42 

do.  to  sq.  inches,  digit  table,  44 

chain,  43 

decimeter,  42 

foot,  42 

do.  to  sq.  meters,  digit  table,  44 

inch,  42 

do.  to  sq.  cm.,  digit  table,  44 

ken,  Japan,  44 

kilometer,  43 

meter,  43 

do.  to  sq.  feet,  digit  table,  44 

do,  to  sq.  yds.,  digit  table,  44 

mil,  41 

mile,  43 

millimeter,  41 

miriameter,  44 


INDEX. 


193 


Square : 

myriameter,  44 

perch,  43 

pole,  43 

rod,  43 

yard,  42 

do.  to  sq.  meters,  digit  table,  44 
Stadium,  ancient,  34 
Stajo,    Corsica,    Leghorn,    Naples, 

Venice,  55 
Standard  candles,  table,  146 

do.  defined,  145 
Standard  cells,  108 

do.,  temperature  corrections,  111 
Standard  electric  lamp,  145 
Standard  time,  93 
Standards,  length,  29 

volume,  45 

weight,  56 

see  also  U.  S.  standards 
Stange,  Sweden,  33 
Starelli,  Sardinia,  55 
Stari,  Austria,  Florence,  55 
Statute  mile  (see  Mile),  31 
Steradian,  89 
Stere,  54 

Stone,  British,  60 
Stoof.  Russia,  55 

Stoop,   Amsterdam,  Antwerp,  Swe- 
den, 55 

Stored  energy,  electricalv  123 
Strength  of  pole,  137 

do.,  physical,  20,  21 
Stress,  electric,  108 
Stress  per  unit  area,  63 
Struck  bushel,  53 
Stubgen,  Germany,  54 
Sucre,  Ecuador,  164 
Suerte,  Uruguay,  173 
Sun,  Japan, 34,  172 
Suppressed  factor,  12,  13,  14,  25,  26 
Surfaces,  41 

digit  tables,  44 

physical,  18 

British  to  U.  S.,  41 

foreign,  44 

and  forces,  63 

and  weights,  63 

density,  22,  24 

tension,  62 

physical,  18 
Susceptance,  105 

physical,  22,  24 
Susceptibility,  132 

physical,  20.  21 
Swedish  lengths,  33 
Swiss  lengths,  33 
Symbols,  text,  28 

tables,  ix 

physical  quantities,  18 
Synodic  month,  94 
System,  absolute,  11 

C.G.S.,  11 

Systeme*  ancien,  French.  33,  54  61 
Systeme,  usuel,  French,  33,  55,  61 


Tables : 

conversion  factors,  tables,  30-178 

do.,  digit  tables,  see  Digit 

do.,  text,  27 

physical  quantities,  17 
Tabriz,  Persia,  173 
Tael,,  China,  164,  166 

Cochin-China,  172 
Tan,  Japan,  44,  172 
Tangent,  grades,  91,  92 
Tarrie,  Algiers,  55 
Telegraph  line  measures,  34 
Temoli,  Naples,  55 
Temperature,  26 

physical,  18 

C9rrections,  standard  cells,  111 

dimensions  of,  13,  26 

scales,  150-163 
Ten  to  the  nth  power,  8 
Tension,  film,  62 
Terze,  31 

Testoon,  Portugal,  165 
Thermal  quantities  (physical),  26 
Thermal  unit,  75 

do.  defined,  74 

do.  to  foot-pounds,  dig.  tab.,  77 

do.  per  minute,  81 
Thermoelectric  height,  23,  25 
Thermometer  scales,  150-163 
Thomson's  law,  128 
Tierce,  53 
Time: 

tables,  94 

text,  93 

physical,  18 

mean  solar,  defined,  93 

sidereal,  defined,  93 

standard,  railway,  93 

and  volume  (discharges),  95 

constant  (electric),  120 

do.,  physical,  23,  25 
To,  Japan,  55,  172 
Toende,  Copenhagen,  55 
Toesa,  Spain,  34 
Toise',  54 

Trance,  33 
Tomans,  Persia,  165 
Tomines,  Spain,  61 
Ton: 

tables,  58 

text,  57 

bloom,  60 

gross,  see  Ton,  long 

do.,  displacement  of  water,  54 

long  or  gross,  58 

do.  to  metric  tons,  digit  table,  59 

do.  vs.  short,  57 

do.,  volume  of  water,  71 

do.,  per  cubic  yard,  69 

do.»  per  square  foot,  66 

do.,  per  square  inch,  67 
metric,  58 

do.  to  long  tons,  dy?it  table,  59 

do.  to  short  tons,  digit  table,  59 
do.,  volume  of  water,  71 
do  ,  per  cubic  meter,  69 


194 


INDEX. 


Ton,  metric,  per  sq.  centimeter,  67 

do.,  per  year,  127 

do.,  kilometer,  78 

net,  see  Ton,  short 

register,  54 

shipping,  54 

short  or  net,  58 

do.  to  metric  tons,  digit  table,  59 

do.,  volume  of  water,  71 

do.,  per  cubic  yard,  69 

do.,  per  mile,  78 

do.,  per  square  foot,  66 

do.,  per  square  inch,  66 

do.,  per  year,  127 

do.,  -mile,  78 
Ton-kilometer,  78 
Ton-mile,  78 
Tonde,  Denmark,  172 
Tondeland,  Denmark,  172 
Tonelada,  Spain.  61 
Tonne,  58 
Tonneau,  58 
Torque,  tables,  74,  78 

defined, 72 

physical,  19 

units,  78 

vs.  energy,  13,  72,  78 
Tortuosity,  18 
Township,  44 
Traction  coefficient,  78 
Traction  energy,  78 
Traction  resistance,  78 
Tractive  effort,  78 
Tractive  force,  78 
Tropical  year,  95 
Troy  weights,  57,  59 

do.  defined,  57 
True  ampere,  113 

defined,  112 
True  coulomb,  116 
True  ohm,  99 

defined,  98 
True  volt,  110 

defined,  109 
Tscharky,  Russia,  55 
Tschetvertaks>  Russia,  165 
Tschetwerik,  Russia,  55 
Tschetwerka,  Russia,  55 
Tschetwert,  Russia,  33,  55 
Tsubo,  Japan,  44,  172 
Tsun,  China,  172 
Turn,  Sweden,  33 
Turn,  cubic,  Sweden,  55 
Tun,  53 

Tunna,  Sweden,  55,  173 
Tunne,  Germany,  54 

Sweden,  55 
Tunnland,  Sweden,  44 


U 

Uncia,  ancient,  34 
Unge,  Switzerland,  61 
Unit,  circular,  defined,  41 
Unit  current-turn,  133 

do.  per  centimeter,  137 
Unit  pole,  magnetic,  138 
Unit-pole-centimeter  unit,  142 
Units,  see  under  respective  names 

absolute  system,  11 

absolute  vs.  concrete,  12 

C.G.S.  system,  11 

changing  —  in  formulas,  6 

concrete  vs.  absolute,  12 

compound  names  of,  4 

inter-relation  of,  1 

three  groups  of,  1 

vs.  quantities,  4 
U.S.  standards,  29,  45,  56,  98,  108 

112,115,117,119,122,124,166 
U.S.  to  British,  volumes,  46 
Unze,  Germany,  60 
Useful  numbers,  170 


V 

Valency,  125,  126,  129 
Vara,  California,  3-1 

Argentine,   Cent.  America,  Chile, 
Cuba,  Curagoa,  Mexico,  172 

Paraguay,  Peru,  Venezuela,  173 

Spain.  34,  173 
Varying  functions,  97 
Vector  potential,  22,  24 
Vector  quantities   96 
Vedro,  Russia,  173 
Velocity  i 

angular,  86 

do-,  physical,  19,  23 

do.,  frequency,  121 

do.,  rate  of  increase  of ,  88 

concrete  units,  86 

light,  86 

do.  as  a  relation,  11,  14,  21,  24, 
25,96 

linear,  85 

do.,  physical,  19 

do.,  rate  of  increase  of,  87 

molecules,  86 
Velte,  France,  54 
Venezolano,  Venezuela,  166 
Vergees,  Isle  of  Jersey,  172 
Verst,  Russia,  33 
Vierling,  Germany,  60 
Viertel,  Antwerp,  55 
Violle.  146 

defined,  145 
Vis-viva,  72 

do.,  physical,  19 


INDEX. 


195 


Vlocka,  Russian  Poland,  173 
Volt: 

tables,  109 

text,  108 

applied,  110 

electro-chemical  energy,  129 

induced,  110 

international,  110 

do.,  denned,  108 

legal,  109 

Reichsanstalt,  110 

relations  to  other  units,  110 

standards,  denned,  108 

true,  110 

do.,  denned,  109 

to  calories,  129 

-ampere,  80 

-coulomb  (joule),  74 
Voltage,  108 

of  decomposition,  128 

do.,  calculation  of,  129 
Vorktum,  Sweden,  33 
Volume: 

table,  46 

text   45 

fundamental  standards,  45 

digit  conversion  tables,  51 

Ehysical,  18 
jreign,  54 
U.S.  to  British,  46 
water,  69,  71 
and  mass,  67 
and  time,  95 
weights,  67 
from  specific  gravities,  69 


W 

Water,  flow  of,  95 

foot  of,  pressure,  65 

meter  of,  pressure,  65 

pressures  of,  63 

specific  heat  of,  171 

volume  of,  71 

weights,  70 
Watt: 

table,  80,  125 

defined,  124 

relations  to  other  units,  125 

magnetic  power   144 

per  candle,  149 

-hour.  76,  123 

defined,  122 

per  gram,  126 

per  minute   81 

per  second,  82 

-second  (joule),  74 
Wave-length   31 
Waves,  periodicity,  86 


Weber,  138 
Weddras,  Russia,  55 
Wedro,  Russia.  55 
Week,  94 
Weight: 

table,  57 

text,  56 

fundamental  standards,  56 

digit  conversion  tables,  59 

physical,  18 

bars,  62 

coatings,  63.  126 

forces,  83 

deposits,  63,  126 

foreign,  60,  172 

materials,  67 

rails,  62 

relative,  chemical,  60 

sheets,  63 

water,  69-71 

wires,  62 

and  length,  62 

and  measures,  tables,  30-173 

do.,  text,  27 

and  surface,  63 

and  money,  167 

and  volume,  67 

from  specific  gravities,  69 
Werschock,  Russia,  33 
Werst,  Russia,  33 
Weston  cells,  defined,  108 

do.,  voltage  of,  110 

do.,  temperature  correction,  111 
Wey,  54 

Winchester  bushel,  53 
Wires,  weights  of,  62 
Wispel,  Germany,  54 
Work: 

tables,  74 

digit  conversion  table,  77 

text,  72 

electrical,  122 

do.,  units,  74,  123 

magnetic,  143 

physical,  19 

rate  of  doing,  79 


Yard:  table,  30 

to  meters,  digit  table,  39 

cubic,  50 

do.,  to  cb.  meters,  digit  table,  51 

solid,  54 

square,  42 

do.,  to  sq.  met.,  digit  table,  44 
Year  ( solar  ^.  94 

calendar,  civil,  common    Julian, 
lunar   sidereal,  94 

anomalistic ,  Gregorian  legal ,  nat- 
ural, tropical,  95 
Yen,  Japan,  165 


196 


INDEX. 


Zehnling,  Germany   60 
Zent,  Germany,  60 
Zoll,   Austria,    Germany,    Switzer- 
land, 33 
Zorzec,  Poland,  55 


n,  as  an  angle,  89 
n,  useful  functions  of,  169 
10  to  the  nth  power,  8 
%  denned,  7 
%  grades,  90 
°/oo  denned,  8 
Voo  grades,  90 

-  (hyphen)  in  names  of  units,  4 


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